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HIGHER MATHEMATICS 



FOE 



STUDENTS OF CHEMISTKY AND PHYSICS 




HIGHER MATHEMATICS 



FOR 



STUDENTS OF CHEMISTRY AND 
PHYSICS 

WITH SPECIAL REFERENCE TO PRACTICAL WORK 



BY 



J. W. M ELL OR, D.Sc. 

LATE SENIOR SCHOLAR AND 1851 EXHIBITION SCHOLAR, 

NEW ZEALAND UNIVERSITY ; RESEARCH FELLOW, 

THE OWENS COLLEGE, MANCHESTER 




LONGMANS, GKEEN, AND CO. 

39 PATERNOSTER ROW, LONDON 

NEW YORK AND BOMBAY 

1902 

All /-fi 



c 



37 



\ 



PREFACE. 

IT is almost impossible to follow the later developments of 
physical or general chemistry without a working knowledge 
of higher mathematics. I have found that the regular 
textbooks of mathematics rather perplex than assist the 
chemical student who seeks a short road to this knowledge, 
for it is not easy to discover the relation which the pure 
abstractions of formal mathematics bear to the problems 
which every day confront the student of Nature's laws, 
and realize the complementary character of mathematical 
and physical processes. 

During the last five years I have taken note of the 
chief difficulties met with in the application of the mathe- 
matician's x and y to physical chemistry, and, as these notes 
have grown, I have sought to make clear how experimental 
results lend themselves to mathematical treatment. I have 
found by trial that it is possible to interest chemical students 
and to give them a working knowledge of mathematics 
by manipulating the results of physical or chemical ob- 
servations. 

I should have hesitated to proceed beyond this experi- 
mental stage if I had not found at The Owens College a 



viii PREFACE. 

set of students eagerly pursuing work in different branches 
of physical chemistry, and most of them looking for help 
in the discussion of their results. When I told my plan 
to the Professor of Chemistry he encouraged me to write 
this book. It has been my aim to carry out his suggestion, 
so I quote his letter as giving the spirit of the book, 
which I only wish I could have carried out to the letter. 

"THE OWENS COLLEGE, 
" MANCHESTER. 

" MY DEAR MELLOR, 

"If you will convert your ideas into words and write a 
book explaining the inwardness of mathematical operations as applied 
to chemical results, I believe you will confer a benefit on many students 
of chemistry. We chemists, as a tribe, fight shy of any symbols 
but our own. I know very well you have the power of winning new 
results in chemistry and discussing them mathematically. Can you 
lead us up the high hill by gentle slopes? Talk to us chemically to 
beguile the way ? Dose us, if need be, ' with learning put lightly, like 
powder hi jam ' ? If you feel you have it in you to lead the way we 
will try to follow, and perhaps some of the youngest of us may succeed^ 
Wouldn't this be a triumph worth working for ? Try. 

" Yours very truly, 

" H. B. DIXON." 
THE OWENS COLLEGE, 
MANCHESTER, May, 1901. 



CONTENTS. 



PAKT I. ELEMENTAEY. 



CHAPTER I. THE DIFFERENTIAL CALCULUS. 

SECTION PAGE 

1. On the Nature of Mathematical Reasoning ..... 1 

2. The Differential Coefficient 4 

3. Differentials 7 

4. Orders of Magnitude . 8 

5. Zero and Infinity 9 

6. Limiting Values 10 

7. The Differential Coefficient of a Differential Coefficient . . .13 

8. Notation 14 

9. Functions 15 

10. Differentiation 17 

11. Is Differentiation a Method of Approximation only ?. ... 19 

12. The Differentiation of Algebraic Functions 22 

13. The Gas Equations of Boyle and van der Waals .... 30 

14. The Differentiation of Trignometrical Functions .... 31 

15. The Differentiation of Inverse Trignometrical Functions. The 

Differentiation of Angles 33 

16. Logarithms and their Differentiation ....... 34 

17. The Differential Coefficient of Exponential Functions . . .38 

18. The " Compound Interest Law " in Nature ..... 39 

19. Successive Differentiation ......... 47 

20. Leibnitz' Theorem 49 

21. Partial Differentiation 50 

22. Euler's Theorem on Homogeneous Functions 56 

23. Successive Partial Differentiation 57 

24. Exact Differentials 57 

25. Integrating Factors 58 

26. Illustrations fr6m Thermodynamics 



x CONTENTS. 

CHAPTER II. COORDINATE OR ANALYTICAL GEOMETRY. 

SECTION PAGE 

27. Cartesian Coordinates .......... 63 

28. Graphical Representation 65 

29. Practical Illustrations of Graphical Representation . ... .66 

30. General Equations of the Straight Line 68 

31. Differential Coefficient of a Point moving on a Straight Line . . 71 

32. Straight Lines Satisfying Conditions 72 

33. Changing the Coordinate Axes ......... 74 

34. The Circle and its Equation - 75 

35. The Parabola and its Equation , . . . . . .76 

36. The Ellipse and its Equation . 78 

37. The Hyperbola and its Equation 80 

38. A Study of Curves . . . .82 

39. The Parabola (resumed) . 85 

40. The Ellipse (resumed) . . . ' . . . . ., . .86 

41. The Hyperbola (resumed) 87 

42. The Rectangular or Equilateral Hyperbola ...... 88 

43. Illustrations of Hyperbolic Curves ... . . . . 89 

44. Polar Coordinates 93 

45. Logarithmic or Equiangular Spiral 95 

46. Trilinear Coordinates and Triangular Diagrams . . . . 97 

47. Orders of Curves .99 

48. Coordinate Geometry in Three Dimensions. Geometry in Space . 101 

49. Orders of Surfaces 110 

50. Periodic or Harmonic Motion . Ill 

51. Generalised Forces and Coordinates . 115 



CHAPTER III. -FUNCTIONS WITH SINGULAR PROPERTIES. 

52. Continuous and Discontinuous Functions 118 

53. Discontinuity accompanied by " Breaks " ...... 119 

54. The Existence of Hydrates in Solution 120 

55. Discontinuity accompanied by Change of Direction .... 124 

56. Maximum and Minimum Values of a Function 129 

57. How to find Maximum and Minimum Values of a Function . . 130 

58. Points of Inflection 132 

59. How to find whether a Curve is Concave or Convex with respect to 

thez-Axis 133 

60. How to find Points of Inflection 134 

61. Multiple Points 135 

62. Cusps 136 

63. Conjugate or Isolated Point 137 

64. Asymptotes 137 

65. Summary 139 

66. Curvature 139 

67. Envelopes 142 

68. Six Problems in Maxima and Minima .... 144 



CONTENTS. xi 
CHAPTER IV. THE INTEGRAL CALCULUS. 

>K TK'N PAGE 

69. Integration 150 

70. Table of Standard Integrals 157 

71. The Simpler Methods of Integration 158 

72. How to find a Value for the Integration Constant .... 162 

73. Integration by the Substitution of a New Variable .... 164 

74. Integration by Parts 168 

75. Integration by Successive Reduction 169 

76. Reduction Formulae (for reference) .... ... 170 

77. Integration by Resolution into Partial Fractions . . . .171 

78. Areas Enclosed by Curves. To Evaluate Definite Integrals . . 177 

79. Graphic Representation of Work 182 

80. Integration between Limits. Definite Integrals .... 183 

81. To find the Length of any Curve 186 

82. Elliptic Integrals 188 

83. The Gamma Function 190 

84. Numerical Table of the Gamma Function . 191 

85. To find the Area of a Surface of Revolution 192 

86. To find the Volume of a Solid of Revolution 193 

87. Successive Integration. Multiple Integrals 194 

88. The Velocity of Chemical Reactions 197 

89. Chemical Equilibrium Incomplete or Reversible Reactions . . 203 

90. Fractional Precipitation 207 

91. The Isothermal Expansion of Gases 208 

92. The Adiabatic Expansion of Gases 211 

93. The Influence of Temperature on Chemical and Physical Changes 

van't Hoffs Formula . .... 214 



CHAPTER V. INFINITE SERIES AND THEIR USES. 

94. What is an Infinite Series ? 218 

95. Soret's Diffusion Experiments 220 

96. Approximate Calculation by Means of Infinite Series . . . 222 

97. Maclaurin's Theorem 226 

98. Useful Deductions from Maclaurin's Theorem 228 

99. Taylor's Theorem 231 

100. The Contact of Curves 235 

101. Extension of Taylor's Theorem 236 

102. The Determination of Maximum and Minimum Values of a Func- 

tion by means of Taylor's Series 237 

103. Indeterminate Functions 242 

104. " The Calculus of Finite Differences " 246 

105. Interpolation and Empirical Formulae 249 

106. To Evaluate the Constants in Empirical or Theoretical Formulae . 255 

107. Approximate Integration 263 

108. Integration by Infinite Series ........ 267 



xii CONTENTS. 

PAET II. ADVANCED. 
CHAPTER VI. HYPERBOLIC FUNCTIONS. 

SECTION PAGE 

109. Euler's Exponential Values of the Sine and Cosine . . ' . . 271 

110. The Derivation of Hyperbolic Functions 272 

111. The Graphic Representation of the Hyperbolic Functions . . 274 

112. Transformation and Conversion Formulae 276 

113. Inverse Hyperbolic Functions 277 

114. Differentiation and Integration of the Hyperbolic Functions . . 277 

115. Demoivre's Theorem ........... 280 

116. Numerical Values of the Hyperbolic Sines and Cosines . . . 280 



CHAPTER VII. HOW TO SOLVE DIFFERENTIAL EQUATIONS. 

117. The Solution of a Differential Equation by the Separation of the 

Variables 283 

118. What is a Differential Equation ? .286 

119. Exact Differential Equations of the First Order .... 289 

120. How to find Integrating Factors . . . . . . . 292 

121. The First Law of Thermodynamics 295 

122. Linear Differential Equations of the First Order .... 296 

123. Differential Equations of the First Order and of the First or 

Higher Degree. Solution by Differentiation . 298 

124. Clairaut's Equation 300 

125. Singular Solutions . . 301 

126. Trajectories ........... 304 

127. Symbols of Operation 304 

128. The Linear Equation of the nth Order 305 

129. The Linear Equation with Constant Coefficients . . . .307 

130. How to find Particular Integrals 310 

131. The Linear Equation with Variable Coefficients .... 315 

132. The Exact Linear Differential Equation 317 

133. The Integration of Equations with Missing Terms .... 319 

134. Equations of Motion, chiefly Oscillatory Motion .... 322 

135. The Velocity of Simultaneous and Dependent Chemical Reactions . 330 

136. Simultaneous Differential Equations 336 

137. Partial Differential Equations 339 

138. What is the Solution of a Partial Differential Equation ? . . 341 

139. The Solution of Partial Differential Equations of the First Order . 344 

140. Partial Differential Equations of the nth Order . . . .346 

141. Linear Partial Equations with Constant Coefficients . . . 347 

142. The Particular Integral of Linear Partial Equations . . .351 

143. The Linear Partial Equation with Variable Coefficients . . . 354 

144. The Integration of Differential Equations in Series .... 355 

145. Harmonic Analysis 357 



CONTENTS. xiii 
CHAPTER VIII. FOURIER'S THEOREM. 

SECTION PAGE 

146. Fourier's Series 360 

147. Evaluation of the Constants in Fourier's Theorem .... 361 

148. The Development of a Function in a Trignometrical Series . . 362 

149. Extension of Fourier's Series 366 

150. Different Forms of Fourier's Integral 368 

151. The Convergency of Fourier's Series 369 

152. The Superposition of Particular Solutions to Satisfy given Con- 

ditions 370 

153. Fourier's Linear Diffusion Law 374 

154. The Solution of Fick's Equation in terms of a Fourier's Series . 376 



PART III. USEFUL RESULTS FROM ALGEBRA 
AND TRIGNOMETRY. 

CHAPTER IX. HOW TO SOLVE NUMERICAL EQUATIONS. 

155. Some General Properties of the Roots of Equations .... 385 

156. The General Solution of Quadratic Equations 387 

157. Graphic Methods for the Approximate Solution of Numerical 

Equations 388 

158. Newton's Method for the Approximate Solution of Numerical 

Equations 390 

159. How to Separate Equal Roots from an Equation .... 391 

160. Sturm's Method of Locating the Real and Unequal Roots of a 

Numerical Equation 392 

161. Homer's Method for Approximating to the Real Roots of Numerical 

Equations 395 

162. van der Waals' Equation of State 398 

CHAPTER X. DETERMINANTS. 

163. Simultaneous Equations 402 

164. The Expansion of Determinants 405 

165. The Solution of Simultaneous Equations 406 

166. Elimination 407 

167. Fundamental Properties of Determinants 407 

168. The Multiplication of Determinants 410 

169. The Differentiation of Determinants 411 

170. Jacobians and Hessians 412 

171. Some Thermodynamic Relations 414 



xiv CONTENTS. 

CHAPTER XI. PROBABILITY AND THE THEORY OF ERRORS. 

SECTION PAGBT 

172. Probability 416 

173. Application to the Kinetic Theory of Gases 421 

174. Errors of Observation 42& 

175. The " Law " of Errors 428 

176. The Probability Integral 431 

177. The Best Representative Value for a Set of Observations . . 433 

178. The Probable Error . . . . . . . . . . . 436- 

179. Mean and Average Errors ......... 439 

180. Numerical Values of the Probability Integrals .... 445 

181. Maxwell's Law of Distribution of Molecular Velocities . . . 448 

182. Constant Errors 451 

183. Proportional Errors ........... 453 

184. Observations of Different Degrees of Accuracy 462 

185. Observations Limited by Conditions ....... 469 

186. Gauss' Method of Solving a Set of Linear Observation Equations . 471 

187. When to Reject Suspected Observations 475 



CHAPTER XII. COLLECTION OF FORMULAE FOR REFERENCE. 

188. Law of Indices and Logarithms ....... 479 

189. Approximate Calculations in Scientific Work . . . . 483 

190. Variation 487 

191. Permutations and Combinations . 489 

192. Mensuration Formulae . . . ~. 490 

193. Bayer's " Strain Theory " of Carbon Ring Compounds . . . 492 

194. Plane Trignometry 493 

195. Spherical Trignometry 501 

196. Summary of Relations among the Hyperbolic Functions . . 504 



CHAPTER XIII. REFERENCE TABLES. 

TABLE 

I. Standard Integrals . . 158 

II. Numerical Values of the Gamma Function .... 507 

III. Standard Integrals (Hyperbolic Functions) .... 278, 506 

IV. Numerical Values of the Hyperbolic Sines 510 

V. Numerical Values of the Hyperbolic Cosines .... 511 

0-6745 
VI. Numerical Values of the Factor , . . . . 512. 

\'n - 1 

0-6745 

VII. Numerical Values of the Factor . .... 512 

N/(n - 1) 

VIII. Numerical Values of the Factor ' 8458 513 

v n(n - I) 

IX. Numerical Values of the Factor - , 513 

n vn - 1 



CONTENTS, 



X. Numerical Values of the Probability Integral c d(hx] 514 



XL Numerical Values of the Probability Integral , r < / <1 ( - ) 515 

XII. Application of Cliauvenet's Criterion 516 

XIII. Signs of the Triguometrical Ratios 495 

XIV. Numerical Values of some Trignometrical Ratios . . . 497 
XV. Squares of Numbers from 10 to 99 516 

XVI. Square Roots of Numbers from O'l to 9'9 ..... 517 

XVII. Square Roots of Numbers from 10 to 100 517 

XVIII. Cube Roots of Numbers from 10 to 100 517 

XIX. Cube Roots of Numbers from 1 to 100 518 

XX. Reciprocals of Numbers from 1 to 100 518 

XXI. Numerical Values of e* from ,r = to .1- = 10 . . . . 518 

XXII. Numerical Values of e~* from .r = to jc = 10 . . . . 519 

o 2 

XXIII. Numerical Values of c- l ~ and e~* from .r = O'l to ^ 5'0 . . 519 

XXIV. Logarithms of Numbers to the Base c 520 

MISCELLANEOUS EXAMPLES ......... 523 

, 527 



" The first thing to be attended to in reading any algebraic treatise is the 
gaining a perfect understanding of the different processes there exhibited, 
and of their connection with one another. This cannot be attained by a 
onere reading of the book, however great the attention which may be given. 
It is impossible in a mathematical work to fill up every process in the 
manner in which it must be filled up in the mind of the student before 
he can be said to have completely mastered it. Many results must be given 
of which the details are suppressed, such are the additions, multiplications-, 
extractions of square root, etc., with which the investigations abound. 
These must not be taken on trust by the student, but must be worked by 
his own pen, which must never be out of his hand while engaged in any 
algebraical process." DE MORGAN, On the Study and Difficulties of Mathe- 
matics, 1831. 



PROLOGUE. 

WHEN Sir Isaac Newton communicated the manuscript of his 
" Methodus fluxionem " to his friends in 1669 he furnished 
science with its most powerful and subtle instrument of 
research. The states and conditions of matter, as they 
occur in Nature, are in a state of perpetual flux, and these 
qualities may be effectively studied by the Newtonian method 
whenever they can be referred to number or subjected to 
measurement (real or imaginary). By the aid of Newton's 
calculus the mode of action of natural changes from moment 
to moment can be portrayed as faithfully as these words 
represent the thoughts at present in my mind. From this, 
the law which controls the whole process can be determined 
with unmistakable certainty. by pure calculation the so- 
called Higher Mathematics. 

This work starts from the thesis that so far as the 
investigator is concerned, 

Higher Mathematics is the art of reasoning about the 
numerical relations between natural phenomena; and the 
several sections of Higher Mathematics are different modes 
of viewing these relations.* 

For instance, I have assumed that the purpose of the 
Differential Calculus is to inquire how natural phenomena 



* In the new (Jermaii .\itmi/>'/i <!<', Naturphttosophie, 1, f>0, 1^02, Ostwald main- 
tains that mathematics is only a language in which the results of experiments may t><- 
conveniently expressed ; and from this standpoint criticises Kant's Metaphysical 
Foundations of Science. '.'/'. footnote, page 1. 



xviii PROLOGUE. 

change from moment to moment. This change may be 
uniform and simple (Chapter I.) ; or it may be associated 
with certain so-called "singularities" (Chapter III.). The 
Integral Calculus (Chapters IV. and VII.) attempts to 
deduce the fundamental principle governing the whole 
course of any natural process from the law regulating the 
momentary states. Coordinate Geometry (Chapter II.) is 
concerned with the study of natural processes by means of 
" pictures " or geometrical figures. Infinite Series (Chapters 
V. and VIII.) furnish approximate ideas about natural pro- 
cesses when other attempts fail. From this, then, we 
proceed to study the various methods (" mathematical 
tools") to be employed in Higher Mathematics. 

This limitation of the scope of Higher Mathematics 
enables us to dispense with many of the formal proofs of 
rules and principles. Much of Sidgwick's * trenchant indict- 
ment of the educational value of formal logic might be urged 
against the subtle formalities which prevail in " school " 
mathematics. While none but logical reasoning could be 
for a moment tolerated, yet too often "its most frequent 
work is to build a pons asinorum over chasms that shrewd 
people can bestride without such a structure ".f 

So far as the tyro is concerned theoretical demonstrations 
are by no means so convincing as is sometimes supposed. 
It is as necessary to learn to "think in letters" and to 
handle numbers and quantities by their symbols as it is to 
learn to swim or to ride a bicycle. The inutility of " general 
proofs" is an everyday experience to the teacher. The be- 
ginner only acquires confidence by reasoning about something 
which allows him to test whether his results are true or 
false ; he is really convinced only after the principle has 
been verified by actual measurement as in 88, say or by 
arithmetical illustration as in 188, say. " The best of all 
proofs," said Oliver Heaviside in a recent number of the 
Electrician, "is to set out the fact descriptively so that it 
can be seen to be a fact". Remembering also that the 
majority of students are only interested in mathematics so 

* A. Sidgwick, The Use of Wards in Reasoning. (A. & C. Black, London.) 
fO. W. Holmes, The Autocrat of the Breakfast Table. (W. Scott, London.) 



PROLOGUE. xix 

far as it is brought to bear directly on problems connected 
with their own work, I have, especially in Part I., explained 
any troublesome principle or rule in terms of some well- 
known natural process. For example, the meaning of the 
differential coefficient and of a limiting ratio is first explained 
in terms of the velocity of a chemical reaction; the differen- 
tiation of exponential functions leads us to compound interest 
and hence to the " Compound Interest Law" in Nature; 
the general equations of the straight line are deduced from 
solubility curves ; discontinuous functions lead us to discuss 
MendeleefFs work on the existence of hydrates in solutions ; 
Wilhelmy's law of mass action prepares us for a detailed 
study of processes of integration; Harcourt and Esson's 
work introduces the study of simultaneous differential equa- 
tions ; Fourier's series is applied to diffusion phenomena, 
etc., etc, Unfortunately, this plan has caused the work to 
assume more formidable dimensions than if the precise and 
rigorous language of the mathematicians had been retained 
throughout. 

I have sometimes found it convenient to evade a tedious 
demonstration by reference to the " regular textbooks ". In 
such cases, if the student wants to "dig deeper," one of the 
following works, according to subject, will be found sufficient : 
Williamson's Differential Calculus, also the same author's 
Integral Calculus (Longmans, Green, & Co., London); 
Forsyth's Differential Equations (Macmillan & Co., London); 
Johnson's Differential Equation* (Wiley & Son, New York). 

Of course, it is not always advisable to evade proofs in 
this summary way. The fundamental assumptions the so- 
called premises employed in deducing some formulae must 
be carefully checked and clearly understood. However 
correct the reasoning may have been, any limitations intro- 
duced as premises must, of necessity, reappear in the con- 
clusions. The resulting formulae can, in consequence, only 
be applied to data which satisfy the limiting conditions. 
The results deduced in Chapter XI. exemplify, in a forcible 
manner, the perils which attend the indiscriminate applica- 
tion of mathematical formulae to experimental data. Some 



ix PROLOGUE. 

formulae are particularly liable to mislead. The " probable 
error " is one of the greatest sinners in this respect. 

The teaching of mathematics by means of abstract 
problems is a good old practice easily abused. The abuse 
has given rise to a widespread conviction that "mathematics 
is the art of problem solving," or, perhaps, the prejudice 
dates from certain painful reminiscences associated with 
the arithmetic of our school-days. 

Under the heading " Examples " I have collected 
laboratory measurements, well-known formulae, practical 
problems and exercises to illustrate the text immediately 
preceding. A few of the problems are abstract exercises in 
pure mathematics, old friends which have run through 
dozens of textbooks. A great number, however, are based 
upon measurements, etc., recorded in papers in the current 
science journals (Continental, American or British), and are 
reproduced in this connection for the first time. 

It can serve no useful purpose to disguise the fact that a 
certain amount of drilling, nay, even of drudgery, is neces- 
sary in some stages, if mathematics is to be of real use as 
a working tool, and not employed simply for quoting the 
results of others. The proper thing, obviously, is to make 
the beginner feel that he is gaining strength and power 
during the drilling. In order to guide the student along 
the right path, hints and explanations have been appended 
to those exercises which have been found to present any 
difficulty. The subject-matter contains no difficulty which 
has not been mastered by beginners of average ability with- 
out the help of a teacher. 

The student of this w r ork is supposed to possess a work- 
ing knowledge of elementary algebra so far as to be able to 
solve a set of simple simultaneous equations,' and to know 
the meaning of a few trignometrical formulae. If any 
difficulty should arise on this head, it is very possible that 
155, 156, or 188 to 194 will say what is required on 
the subject. I have, indeed, every reason to suppose that 
beginners in the study of Higher Mathematics most fre- 
quently find their ideas on the questions discussed in 



PROLOG1 I 

^ l.ss to 194 have L.T\VM so rusty with neglect -,\ 
require refurbishing. 

I have also assumed that the reader is ;ir<|ii;mite<l with 
the elementary principles of chemistry and physics. Should 
any illustration involve some phenomenon with which he 
is not acquainted, there are two remedies to skip it, or to 
look up some textbook. There is no special reason why the 
student should waste time with illustrations in which ho has 
no interest. 

It will be found necessary to procure a set of mathe- 
matical tables containing the common logarithms of numbers 
and numerical values of the natural and logarithmic trigno- 
metrical ratios. Such sets can be purchased for about 
eighteen pence. The other numerical tables required for 
reference in Higher Mathematics are reproduced in the last 
chapter. 

Where I am consciously indebted to any particular 
authority for ideas, either in the design of a diagram or in 
the writing of the text, I have stated the original source 
so that the student may have the opportunity of consulting 
the original for a fuller and perhaps a more lucid discussion. 

I have great pleasure in thanking my friends for assist- 
ance in reading over the proofs, more especially Mr. W. R. 
Anderson, B.Sc., who has verified a great number of the 
examples from the printed slips, and Mr. L. Bradshaw, 
B.8c., who has* carefully checked all the numerical tables. I 
am also pleased to acknowledge the general excellency of the 
printer's share of the work. 

It is, perhaps, too much to hope that all errors have been 
eliminated from the text, and the writer will be grateful 
when apprised of any which may have escaped his notice. 

J. W. M. 






ADDENDA AND CORRIGENDA. 

P. 70, last sentence in footnote to read : " Gay Lussac says that Charles 
had worked on this subject some years before himself, hence, etc." " See also 
(27) p. 526." 

P. 82, fig. 23, insert " R " as described in the text. 

P. 85, fig. 25, the upper " T" should be " T ". 

P. 112, fig. 51, for "f " read "$". 

P. 113, fig. 52, for " " read " r sin e ". 

P. 189, equation 3, the vinculuin should not extend over " df ". 

P. 203, line 10, for " 5-40 " read " 5-76 ". 

P. 269, line 25, add " see (30) p. 526 ". 

P. 289, line 16, for " third " read " first ". 

P. 378, at end of line 1, insert " 322 ". 



PART I. 

ELEMENTARY. 

CHAPTEE I. 
THE DIFFERENTIAL CALCULUS. 

1. On the Nature of Mathematical Reasoning. 

" The philosopher may be delighted with the extent of his views, the 
artificer with the readiness of his hands, but let the one remember 
that without mechanical performance, refined speculation is an 
empty dream, and the other that without theoretical reasoning, 
dexterity is little more than brute instinct." DR. JOHNSON. 

HERBERT SPENCER has de6ned a law of Nature as a proposition 
stating that a certain uniformity has been observed in the relations 
between certain phenomena. In this sense a law of Nature ex- 
presses a mathematical relation between the phenomena under 
consideration. Every physical law, therefore, can be represented 
in the form of a mathematical equation. One of the chief objects 
of scientific investigation is to find out how one thing depends on 
another, and to express this relationship in the form of a mathe- 
matical equation (symbolic or otherwise) is the experimenter's 
ideal goal.* 

There is in some minds an erroneous notion that the methods 
of higher mathematics are prohibitively difficult. Any difficulty 



* Thus Berthelot, in the preface to his celebrated Essai de M4caniquc Chimiqu* 
our la tkennochemie of 1879, described his work as an attempt to base chemistry 
wholly on those mechanical principles which prevail in various branches of physical 
science. Kant, in the preface to his Metaphysischen Anfangsgruenden der Natur- 
wisMnschaft, has said that in every department of physical science there is only so 
much science, properly so called, as there is mathematics. As a consequence, he 
denied to chemistry the name " science ". But there were no "journals of physical 
chemistry" in his time (1786). 

A 






2 HIGHER MATHEMATICS. 1. 

that might arise is rather due to the complicated nature of the 
phenomena alone. Comte has said in his Philosophie Positive, 
' ' our feeble minds can no longer trace the logical consequences of 
the laws of natural phenomena whenever we attempt to simul- 
taneously include more than two or three essential factors ". * 
In consequence it is generally found expedient to introduce 
" simplifying assumptions" into the mathematical analysis. For 
example, in the theory of solutions we pretend that the dissolved 
substance behaves as if it were an indifferent gas. The kinetic 
theory of gases, thermodynamics, and other branches of applied 
mathematics are full of such assumptions. 

By no process of sound reasoning can a conclusion drawn from 
limited data have more than a limited application. Even when 
the comparison between the observed and calculated results is 
considered satisfactory, the errors of observation may quite obscure 
the imperfections of the formula based on incomplete or simplified 
premises. Given a sufficient number of " if's," there is no end to 
the weaving of " cobwebs of learning admirable for the fineness of 
thread and work, but of 110 substance or profit" (Bacon). The 
only safeguard is to compare the deductions of mathematics with 
observation and experiment " for the very simple reason that they 
.are only deductions, and the premises from which they are made 
may be inaccurate or incomplete. We must remember that we 
cannot get more out of the mathematical mill than we put into it, 
though we may get it in a form infinitely more useful for our 
purpose " (John Hopkinson's James Forrest Lecture, 1894). 

The first clause of this last sentence is often quoted in a 
parrot-like way as an objection to mathematics. Nothing but 
real ignorance as to the nature of mathematical reasoning could 
give rise to such a thought. No process of sound reasoning 
can establish a result not contained in the premises. -j- It is 

* I believe that this is the key to the interpretation of Comte' s strange remarks : 
" Every attempt to employ mathematical methods in the study of chemical questions 
must be considered profoundly irrational and contrary to the spirit of chemistry. . . . 
If mathematical analysis should ever hold a prominent place in chemistry an aber- 
ration which is happily almost impossible it would occasion a rapid and a widespread 
degeneration of that science." (Freely translated from the fourth book of Auguste 
Comte's Philosophie Positive, 1830.) 

f Inductive reasoning is, of course, good guessing, not sound reasoning, but the 
finest results in science have been obtained in this way. Calling the guess a "working 
hypothesis," its consequences are tested by experiment in^fvery conceivable way. For 



i. THE DIFFERENTIAL CALCULUS. 3 

admitted on all sides that any demonstration is vicious if it 
contains in the conclusion anything more than was assumed 
in the premises. Why then is mathematics singled out and 
condemned for possessing the essential attribute of all sound 
reasoning? 

It has been said that no science is established on a firm basis 
unless its generalisations can be expressed in terms of 'number, and 
it is the special province of mathematics to assist the investigator 
in finding numerical relations between phenomena. After experi- 
ment, then mathematics. While a science is in the experimental 
or observational stage, there is little scope for discerning numerical 
relations. It is only after the different workers have " collected 
data" that the mathematician is able to deduce the required 
generalisation. Thus a Maxwell followed Faraday and a Newton 
completed Kepler. 

It must not be supposed, however, that these remarks are 
intended to imply that a law of Nature has ever been repre- 
sented by a mathematical expression with perfect exactness. In 
the best of generalisations, hypothetical conditions invariably 
replace the complex state of things which actually obtains in 
Nature. 

There is a prevailing impression that once a mathematical 
formula has been theoretically deduced, the law, embodied in 
the formula, has been sufficiently demonstrated, provided the 
differences between the "calculated" and the "observed" results 
fall within the limits of experimental error. The important point, 
already emphasized, is quite overlooked, namely, that any discrep- 
ancy between theory and fact is masked by errors of observation. 
With improved instruments, and better methods of measurement, 
more accurate data are from time to time available. The errors of 
observation being thus reduced, the approximate nature of the 
formulae becomes more and more apparent. Ultimately, the dis- 
crepancy between theory and fact becomes too great to be ignored. 
It is then necessary to "go over the fundamentals ". New formulae 
must be obtained embodying less of hypothesis, more of fact. Thus, 
from the first bold guess of an original mind, succeeding genera- 
example, the brilliaut work of Fresnel was the sequel of Young's undulatory theory 
of light, and Hertz's recent work was suggested by Maxwell s electro-magnetic theories. 
J. Thomson's remarkable prediction of the influence of pressure on the melting point 
of ice was experimentally verified by Lord Kelvin. See also pages 42, 156, etc. 



4 HIGHER MATHEMATICS. 2. 

tions progress step by step towards a comprehensive and a complete 
formulation of the several laws of Nature.* 

, I shall proceed at once to explain the nature of the more im- 
portant " tools " used in the application of mathematical processes 
to natural phenomena. 

2. The Differential Coefficient. 

Higher mathematics, in general, deals with magnitudes which 
vary in a continuous manner. In order to render such a process 
susceptible of mathematical treatment the magnitude is supposed 
to change during a series of very short intervals of time. The 
shorter the interval the more uniform the process. This conception 
is of fundamental importance. To illustrate, let us consider the 
chemical reaction denoted by the equation : 

H 2 + Cl. 2 = 2HCI, 

(hydrogen) (chlorine) (hydrogen chloride), 

and suppose that the product of the action hydrogen chloride 
is removed from the sphere of the reaction the moment it is 
formed, f 

If thirty cubic centimetres of hydrogen chloride are formed in 
one minute the reaction proceeds with a velocity of 30 c.c. per 
minute. This statement is not meant to imply that O5 c.c. of 
hydrogen chloride is formed during every second of the time of 
observation, for 0*2 c.c. may have been formed in the first second* 
and O8 c.c. during some other second of time. The fact observed 
is that the mean rate of formation of hydrogen chloride is thirty 
cubic centimetres per minute. 

* Most, if not all, the formulae of physics and chemistry are in the earlier stages 
of such a process of evolution. For example, some exact experiments by Forbes and 
by Tait indicate that Fourier's formula (page 375) for the conduction of heat gives 
somewhat discordant results on account of the inexact simplifying assumption : ' ' the 
quantity of heat passing along a given line is proportional to the rate of change of 
temperature " ; Weber has pointed out that Fick's equation (page 376) for the diffusion 
of salts in solution must be modified to allow for the decreasing diffusivity of the salt 
with increasing concentration ; and finally, van der Waals, Clausius, Rankine, Sarrau, 
etc. , have attempted to correct the simple gas equation : pv = R6, by making certain 
assumptions as to the internal structure of the gas. 

f According to Bunsen and Roscoe these conditions are approximately realised 
when a mixture of hydrogen and chlorine gases is confined over water saturated with 
the two gases, and exposed to a constant source of light The water absorbs the HC1 
as fast as it is formed. 



j$ 2. THE DIFFERENTIAL CALCULUS. 5 

If it were now possible to measure the amount formed in thirty 
seconds of time, and 18 c.c. of hydrogen chloride were obtained 
during that interval, this would be equivalent to a velocity of 
reaction of 36 c.c. per minute. In this case the calculated velocity 
would more accurately represent the actual velocity during the 
time of observation, because there would be less time for the rate 
of formation of this compound to vary during thirty than during 
sixty seconds. 

Suppose further that O25 c.c. of hydrogen chloride were pro- 
duced during an interval of one second, the observer would be 
perfectly certain that he had determined the rate of formation of 
this acid with a greater accuracy than before, because there would 
be less time for any variation to take place. * Following out the 
consequences of this reasoning we are quite sure that if an 
observation could be made of the amount of hydrogen chloride 
formed during one-millionth of a second, the rate of formation of 
the compound at this moment would be still more accurate. 

Using the symbol &x to denote the amount of hydrogen chloride 
formed during the very small interval of time 8t, the quotient &x/&t 
represents the velocity of the chemical reaction during this interval 
of time. If we could measure the amount of substance formed 
during an infinitely short interval of time the true velocity (v) of 
the chemical reaction would be denoted by the equation : 



where dx is a symbol used by mathematicians to represent an 
infinitely small amount of something (in this case of HC1), and 
dt a correspondingly short interval of time. Hence it follows 
that neither of these symbols per se is of any practical value, but 
their quotient stands for a perfectly definite conception, namely, 
the rate of formation of hydrogen chloride during an interval of 
time so small that all possibility of error due to variation of speed 
is eliminated. 

The quotient dx/dt is known as the differential coefficient of 
x with respect to t. The value of x obviously depends on what 
value is assigned to t ; for this reason x is called the dependent 
variable, t the independent variable. The differential coefficient 
of x with respect to t thus measures a velocity. 

Just as the idea of the velocity of a chemical reaction represents 

v 



6 HIGHER MATHEMATICS. 2. 

the amount of substance formed in a given time, so the velocity of 
any motion can be expressed in terms of the differential coefficient 
of a distance with respect to time, be the motion that of a train, 
tramcar, bullet, sound-wave, water in a pipe or an electric current. 
Again, we may represent the differential coefficient of the volume 
of a gas, the length of a rod, or the electro-motive force of a 
galvanic element with respect to temperature to obtain the so- 
called temperature coefficient or coefficient of expansion as the case 
might be. The differential coefficient of the volume of a gas with 
respect to pressure furnishes the so-called coefficient of compression, 
which measures the compressibility of a gas. 

From these and similar illustrations which will occur to the 
reader, it will be evident that the conception called by mathe- 
maticians the differential coefficient is not new. Every one 
consciously or unconsciously uses it whenever a "rate," " speed," 
or a "velocity" is in question. 

NOTE ON VELOCITY. In elementary dynamics, velocity (v) is denned as 
rate of motion, and is measured in terms of the distance (s) traversed in the 
time (t). That is to say, 

distance traversed _ ^ 1nr .. . v = s^ 
time t 

It is specially important for us to start with a clear idea of what is meant 
by the terms " velocity," " rate of motion," etc. 

A train is observed to travel a distance of 60 miles in one hour. We 
cannot therefore say that it has travelled 30 miles during the last half hour, 
nor yet that it will travel 30 miles during the next half hour. On the other 
hand, if the train, at any part of its journey, is going at the rate of a mile a 
minute, we can say that the velocity at that particular moment is 60 miles 
an hour. 

Strictly speaking, it is a physical impossibility to actually measure the 
"velocity at any instant," we must therefore understand by this term, the 
mean or average velocity during a very small interval of time, with the proviso 
tliat we can get as near as we please to the actual " velocity at any instant" by 
taking tlw interval of time sufficiently small. 

We shall soon see that "methods of differentiation" will actually enable 
us to find the velocity or rate of change during an interval of time so small 
that the rate of motion has not time to change. The differential coefficient 
is the only true measure of the velocity at any instant of time. 

It is important to distinguish between the average velocity during any 
given interval of time, and the actual velocity at any instant. 

The term " velocity " not only includes the rate of motion, but also the 
direction of the motion. If we agree to represent the velocity of a train 
travelling southwards to London, positive, a train going northwards to 



S :i. THE DIFFERENTIAL CALCULUS. 7 

Aberdeen would be travelling with a negative velocity. Again, if we con- 
ventionally agree to consider the rate of formation of hydrogen chloride from 
hydrogen and chlorine as a positive velocity, the rate of decomposition of 
hydrogen chloride into chlorine and hydrogen will be a negative velocity. 

It is not necessary, for our present purpose, to enter into refined 
distinctions between rate, speed, and velocity. I shall use these terms 
synonymously. 

The concept velocity need not be associated with bodies. Every one is 
familiar with the terms " the velocity of light," " the velocity of sound," " the 
rate of propagation of an explosion wave," etc. The chemical student will 
soon adapt the idea to such phrases as, "the velocity of chemical action," 
41 the speed of catalysis," "the rate of dissociation," "the velocity of dif- 
fusion," " the rate of evaporation," etc. 

It requires no great mental effort to extend the notion still further. If 
a quantity of heat is added to a substance at a uniform rate, the quantity of 
heat (Q) added per degree rise of temperature (6) corresponds exactly with 
the idea of a distance traversed per second of time. Specific heat, therefore, 
may be represented by the differential coefficient dQjd8. Similarly, the in- 
crease in volume (V) (or length) per degree rise of temperature is represented 
by the differential coefficient dV/d6; the decrease in volume (V) per gram of 
pressure (p), is represented by the ratio - dV/dp, where the negative sign 
signifies that the volume decreases with increase of pressure. 

In the above examples, it has been assumed that unit mass or unit volume 
of substance is operated upon, and therefore the differential coefficients re- 
spectively represent specific heat, coefficient of expansion, coefficient of 
compressibility. If we start with unit mass of substance, the coefficient of 
velocity of a chemical reaction would obviously be dx/dt. (What does this 
measure ? The rate of transformation of unit mass of substance.) 

But velocity is generally changing. The velocity of a falling stone 
gradually increases during its descent, while, if a stone is projected upwards, 
its velocity gradually decreases during its ascent. Instead of using the 
awkward term "the velocity of a velocity," the word "acceleration" is 
employed. If the velocity is increasing at a uniform rate, the acceleration, 
or rate of change of velocity, or rate of change of motion, is evidently 

increase of velocity = acceUr(Mon . f m Ll Jb 
time t 

where V Q and v l respectively denote the velocities at the beginning and end 
of the interval of time under consideration. 

Mathematicians have agreed to represent an increasing velocity with a 
positive sign, a decreasing velocity with a negative sign. If a clock gains one 
second an hour, the acceleration is positive, if it loses one second an hour, the 
acceleration is negative. This discussion is resumed in 7. 

3. Differentials. 

It is sometimes convenient to regard dx and dt, or more generally 
dx and dy, as very small quantities which determine the course of 



8 HIGHER MATHEMATICS. $ 4. 

any particular process under investigation. These small magni- 
tudes are called differentials or infinitesimals .* Differentials may 
be treated like ordinary algebraic magnitudes. The quantity of 
hydrogen chloride formed in the time dt is represented by the 
differential dx. Hence from (1), if dxjdt = v, we may write in the 
language of differentials 

dx = v.dt. 

3. Orders of Magnitude. 

If a small number n be divided into a million parts, each part 
(/i/10 6 ) is so very small that it may for all practical purposes be 
neglected in comparison with n. If we agree to call n a magnitude 
of the first order, the quantity w/10 6 is a magnitude of the second 
order. If one of these parts be again subdivided into a million 
parts, each part (w/10 12 ) is extremely small when compared with 
n, and the quantity w/10 12 is a magnitude of the third order. We 
thus obtain a series of magnitudes of the first, second, and higher 
orders, 

n, n x 10~ 6 , n x 10 ~ 12 , . . ., 

each one of which is negligibly small in comparison with those 
which precede it, and very large relative to those which follow. f 

This idea is of great practical use in the reduction of intricate 
expressions to a simpler form more easily manipulated. It is 
usual to reject magnitudes of a higher order than those under 
investigation when the resulting error is so small that it is out- 
side the limits of the " errors of observation " peculiar to that 
method of investigation. (See 96 and 189.) 

In order to prevent any misconception it might be pointed out 
that "great" and "small" in mathematics, like "hot" and 
"cold" in physics, are purely relative terms. The astronomer 
in calculating interstellar distances comprising millions of miles 
takes no notice of a few thousand miles ; while the physicist dare 
not neglect distances of the order of the ten thousandth of an inch 
in his measurements of the wave length of light. 

A term, therefore, is not to be rejected simply because it seems 

* Some one has defined differentials as small quantities " verging on nothing". 

fNote 10 8 = unity followed by eight cyphers, or 100,000,000. lO" 8 is a decimal 
point followed by seven cyphers and unity, or 10~ 8 = 1/10 8 = O'OOOOOOOl. This nota- 
tion is in general use. 



S ft, THE DIFFERENTIAL CALCULUS. 9 

small in an absolute sense, but only when it appears small in 
comparison with a much larger magnitude, and when an exact 
determination of this small quantity has no appreciable effect on 
the magnitude of the larger. In making up a litre of normal 
oxalic acid solution, the weighing of the 63 grams of acid required 
need not be more accurate than to the tenth of a gram. In many 
forms of analytical work, however, the thousandth of a gram is of 
fundamental importance ; an error of a tenth of a grain would 
stultify the result. 

5. Zero and Infinity. 

The words " infinitely small " were used in the second para- 
graph. It is, of course, impossible to conceive of an infinitely small 
or of an infinitely great magnitude, for if it were possible to retain 
either of these quantities before the mind for a moment, it would 
be just as easy to think of a smaller or a greater as the case might 
be. In mathematical thought the word "infinity" (written oo) 
signifies the properties possessed by a magnitude greater than any 
finite magnitude that can be named. For instance, the greater 
we make the radius of a circle, the more approximately does the 
circumference approach a straight line, until, when the radius is 
made infinitely great, the circumference may, without committing 
any sensible error, be taken to represent a straight line. The con- 
sequences of the above definition of infinity have led to some of 
the most important results of higher mathematics. To sum- 
marize, infinity represents neither the magnitude nor the value 
of any particular quantity. The term " infinity " is simply an 
abbreviation for the property of growing large without limit. E.g., 
"tan 90 = oo " means that as an angle approaches 90, its tan- 
gent grows indefinitely large. Now for the opposite of greatness 
smallness. 

In mathematics two meanings are given to the word " zero ". 
The ordinary meaning of the word implies the total absence of mag- 
nitude (called absolute zero). Nothing remains when the thing 
spoken of or thought about is taken away. If four units be taken 
from four units absolutely nothing remains. There is, however, 
another meaning to be attached to the word different from the 
destruction of the thing itself. If a small number be divided by a 
billion we get a sn^all fraction. If this fraction be raised to the 
billionth power we get a number still more nearly equal to absolute 



10 HIGHER MATHEMATICS. $ 0.. 

zero. By continuing this process as long as we please we con- 
tinually approach, but never actually reach, the absolute zero. 
In this relative sense zero relative zero is defined as " an 
infinitely small " or "a vanishingly small number," or " a number 
smaller than any assignable fraction of unity ". For example, we 
might consider a point as an infinitely small circle or an in- 
finitely short line. To put these ideas tersely, absolute zero 
implies that the thing and all its properties are absent ; relative 
zero implies that however small the thing may be its property of 
growing small without limit is alone retained in the mind. This 
will be more rigorously demonstrated in the next paragraph. 

EXAMPLES. After the reader has verified the following results he will 
understand the special meaning to be attached to the zero and infinity of 
mathematical reasoning. Let n be any finite number, and let "?" denote 
an indeterminate magnitude, that is, one whose exact value cannot be de- 
termined : 

(1) 004-00=00; -oD = ?;ftxd = 0;0xO = ?;nxa>=oo; 
0/0 = ?; n/Q = oo ; Qjn = ; oo/O = oo ; O/oo = ; n/ oo = ; oo/n = 00; O n ; 
1/0" = oo ; = ? ; 1/0 = ?; oo = oo ; I/ oo = ; 00 = ? ; I/ 00 = ? ; n" = oc 
when n > 1*, and n when n < 1 ; 1/71* = when n > 1, and 1/n 06 = oo 
when n < 1 ; l x = ?; 1/1*= ? ; n = 1 ; l/ = 1. The last two results are 
proved in "the theory of indices" of any algebraic textbook. 

(2) Let y 1/(1 - x) and put x 1, then y = oo ; if x< l,y is positive, 
and y is negative when x > 1. Thus a variable magnitude may change its 
sign when it becomes infinite. 

(3) log 1 = ; logO = - oo ; log oo = oo. 

6. Limiting Values. 

(i) The sum of an infinite number of terms may have a finite 
value. Converting J into a decimal fraction we obtain 

^ = O-lllll . . . continued to infinity, 
or = T V + T -b + T Jfrv + ... to infinity, 

that is to say, the sum of an infinite number of terms is equal to i- 
a finite term ! If we were to attempt to perform this summa- 
tion we should find that as long as the number of terms is finite 
we could never actually obtain the result i. 

* The signs of inequality are as follows: "=$=" denotes "is not equal to"; 
">," "is greater than"; " ^j>," "is not greater than"; "<," "is less than " - 
and "<," "is not less than". Seep. 454. 

For " = " read "is equivalent to" or "is identical with". 



S <>. THE DIFFERENTIAL CALCULUS. 1 1 

If we omit all terms after the first, the result is ^ less than ,', ; 
if we omit all terms after the third, the result is 1 OU too little ; 
and if we omit all terms after the sixth, the result is 9 00( ) Ouo 
less than -J-, that is to say, the sum of these terms continually 
approaches but is never actually equal to , as long as the number 
of terms is finite. J- is then said to be the limiting value of the 
sum of this series of terms. 

Again, the perimeter of a polygon inscribed in a circle is less 
than the sum of the arcs of the circle, i.e., less than the circum- 
ference of the circle. 

In figure 1, let the arcs AaB, BbC ... be bisected at a, 6 ... 
Join Aa, aA, Bb, . . . Although the perimeter of the second poly- 
gon is greater than the first, it is still less 
than the circumference of the circle. In a 
similar way, if the arcs of this second poly- 
gon are bisected, we get a third polygon 
whose perimeter approaches yet nearer to 
the circumference of the circle. By continu- 
ing this process, a polygon may be obtained 
as nearly equal to the circumference of a 
circle as we please. The circumference of ^ 

the circle is thus the limiting value of the 

perimeter of an inscribed polygon, when the number of its sides is 
increased indefinitely. 

In general, when a variable magnitude x continually approaches 
nearer and nearer to a constant value n so that x can be made to 
differ from n by a quantity less than any assignable magnitude, n 
is said to be the limiting value of x. 

From page 5, it follows that dxjdt is the limiting value of 
r/8, when t is made less than any finite quantity, however small. 
This is written for brevity 

dx__ Bx 

dt ~ TT' = ar 

in words " dxjdt* is the limiting value of 8x/8t whendU becomes 
zero" (relative zero, i.e., small without limit). This notation is 
frequently employed. 

* Although differential quotients are, in this work, written in the form "dx/dt,"' 
. . . , the student in working through the examples and demonstrations, should 




write -T, -- . . . The former method is used to economise space. 



12 



HIGHER MATHEMATICS. 



6. 



(ii) The value of a limiting ratio depends on the relation be- 
tween the two variables. Strictly speaking, the limiting value of the 
ratio Sx/8t has the form , and as such is indeterminate.* But 
for all practical purposes the differential coefficient dx/dt is to be 
regarded as a fraction or quotient (hence the German " Differential- 
quotient "). The quotient dx/dt may also be called a "rate- 
measurer," because it determines the velocity or rate with which 
one quantity varies when an extremely small variation is given 
to the other. The actual value of the ratio dx/dt depends on the 
relation existing between x and t. 

Consider the following three examples (De Morgan) : 
(1) If the point P move on the circumference of the circle towards a fixed 
point Q (Fig. 2), the arc x will diminish at the same time as the chord y. By 
bringing the point P sufficiently near to Q we obtain an arc and its chord 




FIG. 



FIG. 3. 



FIG. 4. 



each less than any given line, that is, the arc and the chord continually 
approach a ratio of equality. Or, the limiting value of the ratio SyjSx is 
unity. 



(2) If ABC (Fig. 3) be any right-angled triangle such that AB = BC. 
By Pythagoras' theorem or Euclid, i., 47, and vi., 4, 



If a line }&, moving towards A, remains parallel to BC, this proportion will 
remain constant even though each side of the triangle ADE is made less 
than any assignable magnitude, however small. That is 

Sx dx 1 



* Indeterminate, because % may have any numerical value we please. It is not 
difficult to see this, e.g., 

% = 0, because 0x0 = 0; # = 1, because Ox 1 = 0; 
$ = 2, because x 2 = ; % = 15, because x 15 = 0; 
= 999,999, because x 999,999 = 0, etc. ' 



S 7. THE DIFFERENTIAL CALCULUS. IS 

D 

(3) Let ABC be a triangle inscribed in a circle (Fig. 4). Draw AB per- 
pendicular to BC. Then by Euclid, vi., 8f 

BC-.AC = AC:DC = x:y. 

If A approaches C until the chord AC becomes indefinitely small, DC will 
also^ become indefinitely small. The above proportion, however, remains. 
When the ratio BC : AC becomes infinitely great, the ratio of AC to DC will 
also become infinitely great, or 

Sx dx 



It therefore follows at once that although two quantities may 
become infinitely small their limiting ratio may have any finite or 
infinite value whatever. 

7. The Differential Coefficient of a Differential Coefficient. 

It will be evident from 2, that the differential coefficient doe& 
not necessarily measure the absolute rate of increase during the 
whole process of formation of hydrogen chloride, but rather the 
rate of formation of that compound which would occur if the- 
velocity remained during the whole interval the same as it was 
during the extremely short interval of time dt. 

In the same reaction, if the hydrogen chloride had been allowed 
to remain mixed with the other reacting gases, the velocity of the- 
chemical reaction would gradually decrease as the amount of 
hydrogen chloride present increased. In other words, the velo- 
city of the reaction would be retarded. 

If we consider the number of cubic centimetres of hydrogen 
chloride formed per second, the rate of change of the velocity of 
the reaction is evidently the limit of the ratio 8v/Bt. A retardation * 
is equivalent to a negative acceleration. If / denotes the acceler- 
ation, then a retardation must be denoted by / with a negative 
sign, or, 

/= -Lt t = Q =-- 
But from (1) 2, v is equal to dx/dt, and hence 

/-XSA 

* 

* The meaning of the term " acceleration " is explained in elementary dynamics. 
If a body moves with an increasing velocity its motion is said to be accelerated. 
Acceleration means the rate at which the velocity of a body changes. 



14 HIGHER MATHEMATICS. 8. 

which is more conveniently written 



an expression denoting the momentary rate of increase in the 
velocity of the action due to the presence of increasing amounts 
of hydrogen chloride. 

The ratio d' 2 x/dt 2 is called the second differential coefficient 
of x with respect to t. 

Just as the first differential coefficient of x with respect to t 
signifies a "velocity," the second differential coefficient of x with 
respect to t denotes an " acceleration ". 

In order to fix these ideas we shall consider a familiar ex- 
periment, namely, that of a stone falling from a vertical height. 
Observation shows that the velocity of the descending stone is 
changing from moment to moment. The above reasoning still 
holds good. We first find the distance (ds) traversed during any 
infinitely short interval of time (dt), that is 

dsfdt = v. 

We next consider the rate at which the velocity changes from one 
moment to another and obtain 

dv/dt = f. . 

Substituting for v, we obtain the second differential coefficient 

ffis 

a?"'' 

which represents the rate of change of velocity or the acceleration 
at any instant of time. In this particular example the acceleration 
is due to the earth's gravitational force, and g is usually written 
instead of/. 

In a similar way it could be shown that the third differential 
coefficient would represent the rate of change of acceleration from 
moment to moment, and so on for the higher differential coefficients 
d n xldt n , which are seldom, if ever, used in practice. A few words 
on notation. 

8. Notation. 

Strictly speaking the symbols 8x, St . . . should be reserved for small 
finite quantities ; dx, dt . . . have no meaning per se. As a matter of fact the 
symbols dx, dt . . . are constantly used in place of Sx, St. . . . It is perhaps 
needless to remark that 5, d, d? . . . do not denote algebraic magnitudes. 



} ). THE DIFFERENTIAL CALCULUS. 15 

In the ratio ^-^ is a symbol of an operation performed on x, as much as 

the symbols " -r " or "/" denote the operation of division. In the present 

5x 
case tlie operation lias been to find the limiting value of the ratio -^ when St is 

ninth' smaller and smaller witlwut limit ; but we constantly find that dxjdt Is 
used when S.r/5/ is intended. The notation we are using is due to Leibnitz. 
Newton, the discoverer of this calculus, superscribed a small dot over the 
Dependent variable for the first differential coefficient, two dots for the second, 

thus a-, a; ... In special cases, besides dyfdx and y, we may have -?-.(//), dy x , 

d?y / c ( N. -' 

a: y , Cj, x' . . . for the first differential coefficient ; j-g, ?/, ( >- J y, ,v v x" . . . 

for the second differential coefficient ; and so on for the higher coefficients or 
derivatives as they are sometimes called. The operation of finding the value 
of the first differential coefficient of any expression is called differentiation. 
The differential calculus is that branch of mathematics which deals with 
these operations. 

9. Functions. 

If the pressure to which a gas is subject be altered, it is known 
that the volume of the gas changes in a proportional way. The 
two magnitudes, pressure p and volume f, are interdependent. 
Any variation of the one is followed by a corresponding variation 
of the other. In mathematical language this idea is included in 
the word "function " ; v is said to be a function of p. The two 
related magnitudes are called variables. Any magnitude which 
remains invariable during a given operation is called a constant. 

In expressing Boyle's law for perfect gases we write this idea 

thus : 

(dependent variable] = f (independent variable), 
or v = f(p), 

meaning that " v is some function of p". There is, however, no 
particular reason why p was chosen as the independent variable. 
The choice of the dependent variable depends on the conditions of 
the experiment alone. We could here have written 



just as correctly as v = f(p). In actions involving time it is 
customary, though not essential, to regard the latter as the in- 
dependent variable, since time changes in a most uniform and 
independent way. Time is the natural independent variable. 

In the same way the area of a circle is a function of the radius, 



16 HIGHER MATHEMATICS. 9. 

so is the volume of a sphere ; the pressure of a gas is a function 
of the density ; the volume of a gas is a function of the tempera- 
ture ; the amount of substance formed in a chemical reaction is a 
function of the time ; the velocity of an explosion wave is a func- 
tion of the density of the medium ; the boiling point of a liquid is 
a function of the atmospheric pressure ; the resistance of a wire to 
the passage of an electric current is a function of the thickness of 
he wire ; the solubility of a salt is a function of the temperature, 
etc. 

The independent variable may be denoted by x when writing 
in general terms, and the dependent variable by y. The relation 
between these variables is variously denoted by the symbols : 

y = f(x); y = <j>(x); V = F W> y = *(*)'> V = /i(*0 * 
Any one of these expressions means nothing more than that "y i& 
some function of x". 

If x lt y l ; x. 2 , 2/ 2 ; # 3 , y z , . . . are corresponding values of x and 
y, we may have 

y = f( x ) ; y\ = /to) ; y* = /to) 

"Let?/ =/(#)" means "take any equation which will enable 
you to calculate y when the value of x is known." 

The word "function" in mathematical language thus implies 
that for every value of x there is a determinate value of y. If v^ 
and p Q are the corresponding values of the pressure and volume of 
a gas in any given state, v and p their respective values in some 
other state, Boyle's law states that 

pv = p Q v . 
Hence, p = p^/v ; or, v = p v /p. 

The value of p or of v can therefore be determined for any 
assigned value of v or p as the case might be. 

A similar rule applies for all physical changes in which two 
magnitudes simultaneously change their values according to some 
fixed "law. It is quite immaterial, from our present point of view, 
whether or not any mathematical expression for the function f(x) 
is known. For instance, although the pressure of the aqueous 
vapour in any vessel containing water and steam is a function of 
the temperature, the actual form of the expression or function 

* For ". . ." read "etc." or "and so on". 



$ 10. THE DIFFERENTIAL CALCULUS. 17 

showing this relation -is not known; but the laws connecting the 
volume of a gas with its temperature and pressure are known 
expressions Boyle and Gay Lussac's laws. The concept thus 
remains even though it is impossible to assign any rule for cal- 
culating the value of a function. In such cases the corresponding 
values of each variable can only be determined by actual obser- 
vation and measurement. In other words, f(x) is a convenient 
symbol to denote any mathematical expression containing x. 
From pages 5 and 13, since 

*-/(*), 

the differential coefficient dyldx is another function of x, say /(a?), 

dyidx = f(x), or df(x)/dx = f(x). 

Similarly the second derivative, d-y/dx 2 , is another function of x, 
say /, 

df(x)ldx = f'(x) ; d*y/dx* =/ ; d?f(x)ldx* =/ ; 
and so on for the higher differential functions. 

The above investigation may be extended to functions of three 
or more variables. Thus the volume of a gas is a function of the 
pressure and temperature. We have tacitly assumed that the 
temperature was constant in our preceding illustration. If the 
pressure and temperature vary simultaneously, 

v = f(p, 0). 
These ideas will be developed later on. 

It might be pointed out that the methods of the calculus are 
usually applied to changes in which the independent variable 
varies continuously, or is a continuous function of the dependent 
variable ; discontinuous functions when they do arise only occur 
for special values of x. See " Continuity and Discontinuity," 
page 118. 

10. Differentiation. 

Before a knowledge of the instantaneous rate of change, dy/dx, 
can be of any practical use, it is necessary to know the actual 
relation, " law," or " form " of the function connecting the varying 
quantities one with another. ( 69 may now be read.) 

The differential calculus is not directly concerned with the 
establishment of any relation between the quantities themselves, 
but rather with the inquiry into the momentary state of the body. 

B 




18 HIGHER MATHEMATICS. 10. 

This momentary state is symbolised by the differential coefficient, 

which thus conveys to the mind a perfectly clear and definite 

conception altogether apart from any 
numerical or practical application. 

The mechanical operations of finding 
the differential coefficient of one variable 
with respect to another in any expres- 
sion are no more difficult than ordinary 
algebraic processes. Before describing 
the practical methods of differentiation 
it will be instructive to study a geo- 
metrical illustration of the process. 

Let x (Fig. 5) be the side of a square, 

and let there be an increment in the area of the square due to an 

increase of h in the variable x. 

The original area of the square = x 2 

The new area = (x + h) 2 

The increment in the area = (x + h) 2 - x 2 = 2xh + IP . (3) 

This equation is true, whatever value be given to h. The 
smaller the increment h the less does the value of h 2 become. 
If this increment h ultimately become indefinitely small, then h z , 
being of a very high order of magnitude, may be neglected. For 
-example, if when x = 1, 

h = 1, increment in area 2 + 1 ; 
fc = A, ='2 + T ^; 

= -002 + 1 . 00 ;. 000 , etc. 



If, therefore, dy denotes the infinitely small increment in the 
area (y) of the square corresponding to an infinitely small incre- 
ment dx in two adjoining sides (x), then, in the language of 
differentials, 

increment y = Zxh, becomes, dy = %x.dx. 

(See the historical note, page 20.) 

The same result can be deduced by means of limiting ratios. 
For instance, consider the ratio of any increment in the area (y) 
to any increment in the length of a side of the square (x). 

increment y 
h 



11. THE DIFFERENTIAL CALCULUS. 19 

and when the value of h is zero 



EXAMPLES. (1) Show, by similar reasoning to the above, that if the three 
adjoining sides (x) of a cube receive an increment /*, then Lt h = ^ = 3x z . 

(2) Prove that if the radius (r) of a circle be increased by an amount h, 
the increment in the area of the circle will be (2rh + h 2 ) v. Show that the 
limiting ratio (dyjdx) in this case is 2irr. 

The former method of differentiation is known as " Leibnitz's 
method of differentials," the latter, " Newton's method of limits". 
It cannot be denied that while Newton's method is rigorous, 
exact, and satisfying, Leibnitz's at once raises the question : 

11. Is Differentiation a Method of Approximation only? 

The method of differentiation might at first sight be regarded 
as a method of approximation, for these small quantities appear 
to be rejected only because this may be done without committing 
any sensible error. For this reason, in its early days, the calculus 
was subject to much opposition on metaphysical grounds. Bishop 
Berkeley called these limiting ratios " the ghosts of departed quan- 
tities". A little consideration, however, will show that these 
small quantities must be rejected in order that no error may be 
committed in the calculation. The process of elimination is 
essential to the operation. 

Assuming that the quantities under investigation are con- 
tinuous, and noting that the smaller the differentials the closer 
the approximation to absolute accuracy, our reason is satisfied to 
reject the differentials, when they become so small as to be no 
longer perceptible to our senses. The psychological process that 
gives rise to this train of thought leads to the inevitable conclusion 
that this mode of representing the process is the true one. More- 
over, the validity of the reasoning is justified by its results. 

The following remarks on this question are freely translated 
from Carnot's Reflexions sur la Metaphysique du Calcul In- 
finitesimal. " The essential merit, the sublimity, one may say, 
of the infinitesimal (or differential) method lies in the fact that it 
is as easily performed as a simple method of approximation, and 
as accurate as the results of an ordinary calculation. This im- 



20 HIGHER MATHEMATICS. 11. 

mense advantage would be lost, or at any rate greatly diminished, 
if, under the pretence of obtaining a greater degree of accuracy 
throughout the whole process, we were to substitute for the 
simple method given by Leibnitz * one less convenient and less 
in accord with the probable course of the natural event. If this 
method is accurate in its results, as no one doubts at this day ; if 
we always have recourse to it in difficult questions, what need is 
there to supplant it by complicated and indirect means? Why 
content ourselves with founding it on inductions and analogies 
with the results furnished by other means when it can be de- 
monstrated directly and generally, more easily, perhaps, than any 
of these very methods ? 

" The objections which have been raised against it are based 
on the false supposition that the errors made by neglecting in- 
finitesimally small quantities during the actual calculation are 
still to be found in the result of the calculation, however small 
they may be. Now this is not the case. The error is of necessity 
removed from the result by elimination. It is indeed a strange 
thing that every one did not from the very first realise the true 
character of infinitesimal quantities, and see that a conclusive 
answer to all objections lies in this indispensable process of 
elimination." (Paris, p. 215, 1813.) 

HISTORICAL NOTE. The beginner will have noticed that, unlike algebra 
and arithmetic, higher mathematics postulates that number is capable of 
gradual growth. The differential calculus is concerned with the rate at which 
quantities increase or diminish. There are three modes of viewing this 
growth : 

. 1. Leibnitz 1 s "method of infinitesimals or differentials". According to 
this, a quantity is supposed to pass from one degree of magnitude to another 
by the continual addition of infinitely small parts, called infinitesimals or 
differentials. Infinitesimals may have different orders of magnitude. Thus, 
the product dx . dy is an infinitesimal of the second order, infinitely small in 
comparison with the product y . dx, or x . dy. 

In the preceding section ( 10, see also Fig. 6, 12, and Fig. 8, 21) it 
was shown that when each of two sides of a square receives a small increment 
h, the corresponding increment in the area is 2xh + h 2 . When h is made 
indefinitely small and equal to say dx, then (dx)* is vanishingly small in 
comparison with x . dx. Hence, 

dy = 2x . dx. 

* Isaac Newton discovered the fundamental process of the " differential " calculus 
in 1665-69. Leibnitz improved the notation in 1677. Leibnitz is also said to have 
made the discovery independently of Newton. 



11. THE DIFFERENTIAL CALCULUS: 21 

In calculations involving quantities which are ultimately made to ap- 
proach the limit zero, the higher orders of infinitesimals may be rejected at 
any stage of the process. Only the lowest orders of infinitesimals are, as a 
rule, retained. See (5), page 523. 

2. Newton's " metliod of rates or fluxions ". Here, the velocity or rate 
with which the quantity is generated is employed. The measure of this 
velocity is called a fluxion. A fluxion, written .f, y, . . . is equivalent to our 
dxldt,dyjdt, . . . 

These two methods are modifications of one idea. It is all a question of 
notation or definition. While Leibnitz referred the rate of change of a 
dependent variable */, to an independent variable x, Newton referred each 
variable to " uniformly flowing " time. Leibnitz assumed that when x receives 
an increment dx t y is increased by an amount dy. Newton conceived these 
changes to occupy a certain time dt, so that y increases with a velocity y, as 
x increases with a velocity x. This relation may be written symbolically, 

y dy 
dx = xdt, dy = i/dt, and therefore, = -TV 

The method of fluxions is not in general use, perhaps because of its more 
abstruse character. It is occasionally employed in mechanics. 

3. Newton's "method of limits". This has been set forth in 2, 6, et 
seq* The ultimate limiting ratio is considered as a fixed quantity to which 
the ratio of the two variables can be made to approximate as closely as we 
please. 

The methods of limits and of infinitesimals are employed indiscriminately 
in this work, according as the one or the other appeared the more instructive 
or convenient. As a rule, it is easier to represent a process mathematically 
by the method of infinitesimals. The determination of the limiting ratio 
frequently involves more complicated operations than is required by Leibnitz's 
method. (Compare 85, and 86.) 

"The limiting ratio," says Carnot (/. c., p. 210), "is neither more nor less 
difficult to define than an infinitely small quantity. ... To proceed rigor- 
ously by the method of limits it is necessary to lay down the definition of a 
limiting ratio. But this is the definition, or rather, this ought to be the 
definition, of an infinitely small quantity." It follows, therefore, that the 
psychological process of reducing quantities down to their limiting ratios is 
equivalent to the rejection of terms involving the higher orders of infinitesi- 
mals. These operations have been indicated side by side in 10. 

The earlier part of Professor Williamson's article on the "Infinitesimal 
Calculus," in the Encyclopaedia Britannica (9th edit.), contains some interest- 
ing details on the evolution of the calculus. 

We may now take up the routine processes of differentiation. 

* The method of limits is sometimes said to have been suggested by d'Alembert. 
But this sarant has stated positively iu the Encyclopedic Mathematique (1784-1789), 
1'art. " Differentiel," that he has but interpreted the later views of Newton set forth in 
The Principia. 



22 HIGHER MATHEMATICS. 12. 

12. The Differentiation of Algebraic Functions. 

An algebraic function of x is an expression containing terms 
which involve only the operations of addition, subtraction, multi- 
plication, division, evolution (root extraction) or involution. For 
instance, x 2 y + $x + y% - ax = 1 is an algebraic function. Func- 
tions that cannot be so expressed are termed transcendental 
functions. Thus, sin x = y, log x = y, e x = y are transcendental 
functions. 

On page 18 a method was described for finding the differential 
coefficient of y = x' 2 , by the following series of operations : 

(1) Give an arbitrary increment h to x in the original function ; 

(2) subtract the original function x 2 from the new value of 

(x + h) 2 found in (1); 

(3) divide the result of (2) by h the increment of x ; 

(4) find the limiting value of this ratio when h = 0. 

This procedure must be carefully noted ; it lies at the basis of 
all processes of differentiation. In this way it can be shown that 

if y = x 2 , dy/dx 2x, 
if y = z 3 , dy/dx = 3z 2 , 
if y = x 4 , dy/dx 4# 3 , etc. 

(1) To find the differential coefficient of any power of a variable. 

By actual multiplication we shall find that 

(x + 7*) 2 = (x + h) (x + h) = x* + 2hx + li z ; 
(x + lif = (x + h)*(x + h) = x* 



Continuing this process as far as we please, we shall find that 

(x + fc) = a" + rca*- i fe+ n ( n ~ 9 1 )s- 2 fe a + . . . + "ajfe"- 1 + ft". (1) 

This result, known as the binomial theorem, enables us to raise any ex- 
pression of the type x + h to any power of n (where n is positive integer, i.e., 
a positive whole number, not a fraction) without going through the actual 
process of successive multiplication. Exactly the same thing holds for (x - h) n . 

To find the differential coefficient of 

y = x n . 
Let each side of this expression receive a small increment so that 

(y + h') - y = (incr. y) = (x + h) - x n . 
From the binomial theorem, (1) above 

(incr. y) = nx n ~ l h + n(n - l)x n ~ 2 /i 2 + . . . 



12. THE DIFFERENTIAL CALCULUS. 23 

Divide by increment x, namely h. 



. re) 
Hence when /i is made zero 

7 . (tncr. y) _ Lii , (x + fe) - x n _ _ 1 

L '*-(7S^)- w ' A=o s- 

That is to say 

*. 3^ .*"-!. ... (2) 

dx dx 

Hence the rule : to find the differential coefficient of any power 
of x, diminish the index by unity and multiply the power of x so 
obtained by the original exponent (or index). 

EXAMPLES. (1) If y = X K , show that dy/dx = 6x 5 . 

(2) If y = x, show that dy/dx = 20z 19 . 

(3) If y = a(x 5 ), show that dyjdx = a(5x*} = 5ax*. 

(4) If the diameter of a spherical soap bubble increases uniformly at the 
rate of 0-1 centimetre per second, show that the capacity is increasing at the 
rate of O2ir centimetre per second when the diameter becomes 2 centimetres. 
Note : y = ^irD 3 , (23), page 492 ; 

, .-. dy = x TT x 2 2 x 0-1 = 0-2ir. 



(2) To find the differential coefficient of the sum or difference of 
any number of functions. Let u, v, w . . . be functions of x ; y 
their sum. Let u t v ly w lt . . . , y lt be the respective values of 
these functions when x is changed to x + h, then 

y = u + v + w+. . . ; y 1 = MJ + t\ + w-^ + . . . 
Hence y l - y = (^ - u) + (^ - v) + (w 1 - w) + . . . , 
that is (incr. y) = (incr. u) + (incr. v) + (incr. w) + . . . , 
dividing by h 

(incr. y) (incr. u) (incr. v) (incr. w) 

- ^= ~ _|_ _ -|- _|_ . . j 

h h h h 



t h ._ . . 

(incr. x) dx dx dx dx 

If some of the symbols had had a minus instead of a plus sign, 
a corresponding result would have been obtained. For instance, 
if y = u - v - w - . . . , 

then to_ = to _ dv _ to _ (f) 

dx dx dx dx 

The differential coefficient of the sum or difference of any num- 
ber of functions is, therefore, equal to the sum or difference of the 
differential coefficients of the several functions. 



24 HIGHER MATHEMATICS. 12. 

(3) To differentiate a polynomial * raised to any power. Let 

y = (ax + x^ n - 

Eegarding the expression in brackets as one variable raised to the 
power of n, we get 

dy = n (ax + a? 2 )" ~ J d (ax + x 2 ). 
Differentiating the last term, 

^. = n(ax + a?) n - l (a + 2x). . . (5) 

Thus, to find the differential coefficient of a polynomial raised 
to any power, diminish the exponent of the power by unity and 
multiply the expression so obtained by the differential coefficient 
of the polynomial, and the original exponent. 

(4) The differential coefficient of any constant term is zero. 
Since a constant term is essentially a quantity that does not vary, 
if y = c, dy must be absolute zero. Let 

y = x n + C 

then (incr. y) = (x + h) n + c - (x n + c) 



Lth . = 

- (incr. x) dx 

where the constant term has disappeared. 

(5) To find the differential coefficient of the product of a variable 
and a constant quantity. Let 

y = ax n ; (incr. y) = a (x + h) n - ax n ; 



Therefore 



* A polynomial is an expression containing two or more terms connected by plus 
or minus signs. Thus, a + bx ; ax + by + z, etc. A binomial contains two such 
terms. 

t Note I! = l;2! = lx2;3! = lx2x3;w! = lx2x3x . . . x (n - 2) x (n - 1) x n. 
Strictly speaking, 0! has no meaning ; mathematicians, however, find it convenient 
to make 0! = 1. This notation is due to Kramp. " n\ " is read " factorial n ". 



12. THE DIFFERENTIAL CALCULUS. 25 

The differential coefficient of the product of a variable quantity 
and a constant is thus equal to the constant multiplied by the 
differential coefficient of the variable. 

EXAMPLES. Some illustrations of this process have been given in pre- 
ceding examples. 

(1) If y = (1 - u.- 2 ) 3 , show that dy/dx = - 6x (1 - z 2 ) 2 . 

(2) If y = x - 2z 2 , show that dyjdx = 1 - 4z. 

(3) Young's formula for the relation between the vapour pressure p and 
the temperature of isopentane at constant volume is, p = be - a, where a 
and b are empirical constants. Hence show that the ratio of the change of 
pressure with temperature is constant and equal to b. 

(4) Mendelteff's formula for the superficial tension s of a perfect liquid at 
Any temperature 6 is, s = a - bO, where a and b are constants. Hence show 
that rate of change of s with is constant. Ansr. - 6. 

(5) Calendar's formula for the variation of the electrical resistance R of 
a platinum wire with temperature is, B = R (l + o0 + )80 2 ), where a and ft 
are constant. Find the increase in the resistance of the wire for a small rise 
of temperature. Ansr. dR = R Q (a + 200) d6. 

(6) If the volume of a gramme of water varies as 1 + (0 - 4) 2 /144,000, where 
denotes the temperature (C), show that the coefficient of cubical expansion 
of water at any temperature is equal to -000013889 K (0 - 4), where K is the 
constant of proportion (2), page 487. 

(7) A piston slides freely in a circular cylinder (diameter 6 in.). At what 
rate is the piston moving when steam is admitted into the cylinder at the 
rate of 11 cubic feet per second ? 

Let v denote the volume, x the height of the piston at any moment. 
Prom (25), page 492, 

v = v() 2 x ; .-. dv = if(^fdx. - 

But we require the value of dxjdt. Divide the last expression through with 
dt t let IT = -V-, 

(8) If the quantity of heat (Q) necessary to raise the temperature of a gram 
of solid from to is represented by 

Q = a6 + be 2 + C0 3 

^where a, 6, c, are constants), what is the specific heat of the substance at ? 
Hint. Compare the meaning of dQfdO with your definition of specific heat. 
Ansr. a + 260 + 3c0 2 . 

(6) To find the differential coefficient of the product of any 
number of functions. Let 

y = uv 

where u and v are functions of x. When x becomes x + h, u, v 
and y become u lt v v y lt or U Y = u + h, v l = v + h . . . Then 



26 HIGHER MATHEMATICS. 12. 

add and subtract uv l from the second member of this last equation f 
and transpose the terms so that 

or (incr. y) = u(incr. v) + (v + h) (incr. u). 

Divide by h and find the limit when h = 

(incr. y) dv du 
h = (incr. x) dx dx' 

dy d(uv) dv du 

or -f- = j ^ = u-j + v-j . (Q\ 

dx dx dx dx 

In the language of differentials 

dy = d(uv) = udv + vdu. ... (9) 
Similarly, by taking any number of functions, say 

y = uvw. 
Put vw = z then y = uz. 

dv du dz 
From (8) = ^ + u- & - 

Divide through by y or its equivalent uz and 
1 dy 1 du I dz 
y dx u dx z dx 
Substituting vw for z we get 

1 dz _ 1 dv 1 dw 
z dx v dx w dx 
dy du dv dw 

V dx 
and so on for the products of a greater number of terms. 

To find the differential coefficient of any number of terms r 
multiply the differential coefficient of each separate function by 
the product of all the remaining functions and add up all the 
results. 

This may be illustrated by a geometrical figure similar to that 

of page 18. In the rectangle 
(Fig. 6) let the unequal sides 
be denoted by x and y. Let 
x and y be increased by their 
differentials dx and dy. Then 
the increment of the area will 
FIG- 6- be represented by the shaded 

parts, which are in turn represented by the areas of the parallelo- 
grams xdy + ydx + dxdy, but at the limit dxdy vanishes, as. 
previously shown. 



av u,w x-.fy. 

-= = VWT~ + UW-T~ + UV~T~t (*") 

dx dx dx 




12. THE DIFFERENTIAL CALCULUS. 27 

EXAMPLES. (1) Show geometrically that the differential of a small incre- 
ment in the capacity of a rectangular solid figure whose unequal sides are 
x, y, z is denoted by the expression xydz + yzdx + zxdy. Hence, show that 
if an ingot of gold expands uniformly in its linear dimensions at the rate of 
0*001 units per second, its volume (V) is increasing at the rate of dV/dt = O'llO 
units per second, when the dimensions of the ingot are 4 by 5 by 10 units. 

(2) If y = (x - 1) (x - 2) (x - 3), dy/dx = 3z 2 - 12z + 11. 

(3) If y = x*(l + az 2 ) (1 - az 2 ), dyjdx = 1x - 6a?x\ 

(7) To find the differential coefficient of a fraction or quotient. 

u 
Let </ = -> 

where u and v are functions of x. When x becomes x + h, u v 
and y become respectively u^ v l and y v such that u^ = u + h, etc. 
Then y = u^v^ and 



u 



V, - y = 

l 



V l V VjV 

add and subtract u/v r Then divide by h and 
(incr. y) 

-- 



(incr. y) / (incr. u) 

(incr. a?) \ (incr. x) 

(incr. y} / du 




h --. -- 
- (incr. x) \ dx 

/u\ 
or dy_ = 3_ _d^_dx . . . (11) 

dx dx v' 2 

In words, to find the differential coefficient of a fraction or of a 
quotient, subtract the product of the numerator into the differ- 
ential coefficient of the denominator, from the product of the 
denominator into the differential coefficient of the numerator, and 
divide by the square of the denominator. 

In the language of differentials the last result may be written 
in the more useful form : 

. . (12) 



SPECIAL CASE. If the numerator of the fraction be constant, 
say c, then 

y = c/x. 
dy = (xdc - cdx)/x 2 = - cdx/x 2 ; 






28 HIGHER MATHEMATICS. 12. 

EXAMPLES. (1) If y = -r^ show that dyldx = 11(1 - x) z . 

J. fly 

(2) If y = (1 + z 2 )/(l - a; 2 ), show that dy/dx = 4z/(l - a; 2 ) 2 . 

(3) If y = a\x n , show that <fy/<&c = - na\x n + l . 

(4) If y = x*l(x* - 1) - x^Kx - 1), show that dyldx = 2z/(z 2 I) 2 . 

(5) The refractive index (/*) of a ray of light of wave-length A is, according 
to Christoffel's dispersion formula, 

t /* = ^o \/ 2 /{ \/l + A O /A + N/I - A O /A}, 

where /i and A are constants. Find the change in the refractive index 
corresponding to a small change in the wave-length of the light. Ansr. 
dfijd\ = - J u 3 A 2 /{2A 3 Ai 2 V(l - V/A 2 )}- Tt is not often 80 difficult a differentia- 
tion occurs in practice. The most troublesome part of the work is to reduce 

- A /A)}/A 2 

V(i - *o 

to the answer given, by multiplying the numerator and denominator of the 
right member with the proper factors to get ^ 3 . Of course the student is not 
using this abbreviated symbol of division. See footnote, page 11. 

(8) To find the differential coefficient of a function affected with 
any exponent. Since the binomial theorem is true for any ex- 
ponent positive or negative, fractional or integral, formula (2) may 
be regarded as quite general. To illustrate this consider the three 
cases. 

CASE I. When n is a positive integer. It follows directly 

d(x n ) 

--1 = nx n ~ l . 
ax 

CASE II. When n is a positive fraction. Let n p/q, where p 

and q are any integers,, then 

p 

y = X'i- 
Raising each term to the gth power 

y 9 = x*. 

By differentiation, using the notation of differentials 
qy q ~ l dy = px p ~ 1 dx, 



_ 
But since y = x q , 

M-P 

yi-^ = X i ' 

By substituting this value of y 9 ~ * in the preceding result, we 

obtain 

dy px p - l x p/g 
dx ~ ~ x p ' 



12. THE DIFFERENTIAL CALCULUS. 29 

or simplified: ax = ~ X ^~' t .... (14) 

which has exactly the same form as if n were a positive integer. 
CASE III. When n is a negative integer or a negative fraction. 

Let 

y = x ~ w , then y = l/x n . 

Differentiating this as if it were a fraction, (13) above 
dy/dx = - nx n ~ W, 

or, on reduction, ^| = ^g-* = - nx~ n -* 1 . (15) 

Thus the method of page 23 is quite general. 

EXAMPLES. (1) Matthiessen's formula for the variation of the electrical 
resistance R of a platinum wire with temperature 0, between and 100 is 
R = R Q (1 - aQ + b(P) - 1 . Find the increase in the resistance of the wire for 
a small change of temperature. Ansr. dR/dO = R*(a - 2&0)/# . Note a and;6 
are constants. 

(2) Siemens' formula for the relation between the electrical resistance of 
a metallic wire and temperature is, R = R (l + aO + b x/0). Hence find the 
rate of change of resistance with temperature. Ansr. RQ(O, + $b6~ i). 

(9) To find the differential coefficient of a function of a function. 

Let 

y =f(x) and u = <j>(y). 

It is required to find the differential coefficient of u with respect 

to x. Let u and y receive small increments so that y l = y + h 

and MJ = u + h and x 1 = x + h. Then 

U-L - u HI - u U\ ~ y (incr. u) (incr. u) (incr. y) 
x l - x ~ y l - y ' x l - x' or (incr. x) ~ (incr. y) ' (incr. x)' 

which is true, however small the increment may be. At the limit, 

therefore, 

du_du dy 

dx ~ dy ' dx ( ' 

In a similar way if 

u = <j>(w) ; w = ^(y) ; and y = /(or), 

and the differential coefficient of u with respect to x is required 
since 

du du dw dw _ dw dy 
dx = dw ' Ttf and dx = dy ' dx' 

dy du dw dy 
'dx- = fa'd 'dx ' ' ' 



30 HIGHER MATHEMATICS. 13. 



(10) To prove that 4- = 1 --. Since it has just been shown that 

du du dy 
dx dy ' dx 

is true for all values of x, we may assume that when u = x 
dx dy dx !dy 



EXAMPLES. (1) If y = x n /(l + x) n , show that dyjdx = nx n ~ l l(l 4- x) n + l . 

(2) liy= 1/V(1 - a; 2 ), show that dy/dx = x\ ^(1 - x 2 )*. 

(3) Iiy = a+ If *Jx, show that dy/dx = - % ^x-' A . 

(4) If y = a + bx/c, show that dyjdx = bjc. 

The use of formula (16) often simplifies the actual process of differentia- 
tion ; for instance, it is required to differentiate the expression 

(5) u - N /(a 2 - x 2 ). Assume y = a- 2 - x 2 . Then u = \V> y = a 2 - # 2 , and 
dyldx = -x(a? - x*)-$- 

This is an easy example which could be done at sight ; it is given here to 
illustrate the method. 

Every type of algebraic expression has now been investigated, 
and by the application of these principles any algebraic function may 
be differentiated. Before proceeding to transcendental functions 
(that is to say, functions which contain trignometrical, logarithmic 
or other terms not algebraic), it seems a convenient opportunity 
to apply our knowledge to the well-known equations of Boyle and 
van der Waals. These equations will also be discussed from other 
points of view later on. 

13. The Gas Equations of Boyle and van der Waals. 

In van der Waals' equation, at a constant temperature, 

(p + a/v 2 ) (v - b) = constant, . . (1) 
where b is a constant depending on the volume of the molecule, a 
is a constant depending on intermolecular attraction. Differenti- 
ating with respect to p and v t we obtain, as on pages 24 and 25, 

(v - b)d(p + a/v 2 ) + (p + a/v 2 )d(v - b) = 0, 
and therefore 

dv ,.// a 2a6\ 

= -( V -b)(p -- g-H-xV (2) 
dp 7 v v 2 v s ) 

The differential coefficient dv/dp measures the compressibility of 
the gas (page 7). 

If the gas strictly obeyed Boyle's law, a = b = 0, and we should 
have 

dv/dp = - v/p i . . . (3) 



14. THE DIFFERENTIAL CALCULUS. 31 

The negative sign in these equations means that the volume of 
the gas decreases with increase of pressure. Any gas, therefore, 
will be more or less sensitive to changes of pressure than Boyle's 
law indicates, according as the differential coefficient of (2) is 
greater or less than that of (3), that is according as 
(v - b)/(p - alv* + 2ab/v s ) ^ v/p, 
or as pv - pb < pv - a/v 4- 2ab/v 2 , 

or as pb < a/v - 2ab/v' 2 , 

- -. a ( 2a 
or as P v h ' ' ' ' 

If Boyle's law were strictly obeyed 

pv = constant, .... (5) 

but if the gas be less sensitive to pressure than Boyle's law 
indicates, so that, in order to produce a small contraction, the 
pressure has to be increased a little more than Boyle's law 
demands, 

pv increases with increase of pressure ; 

while if the gas be more sensitive to pressure than Boyle's law 
provides for, 

pv decreases with increase of pressure. 

Some valuable deductions as to interinolecular action have been 
drawn by comparing the behaviour of gases under compression in 
the light of equations similar to (4) and (5). For this the reader 
is referred to the proper textbooks. 

But this is not all. From (5), if c = constant, v = cfp, which 
gives on differentiation 

dv/dp = - c/p 2 , 

or the ratio of the decrease in volume to the increase of pressure, 
is inversely as the square of the pressure. By simple substitution 
of p = 2, 3, 4, . . . in the last equation we obtain 

dv/dp = J, i, T V . . . 

when c = unity. In other words, the greater the pressure to 
which a gas is subjected the less the corresponding diminution in 
volume for any subsequent increase of pressure. 

14. The Differentiation of Trignometrical Functions. 

A trignometrical function is any expression containing trigno- 
metrical ratios, sines, cosines, etc. The elementary definitions 
of trignometry are discussed on page 493. We may therefore 



32 HIGHER MATHEMATICS. 14. 

pass at once in medias res. There is no new principle to be 
learned. 

(1) The differential coefficient of sin x is cos x. Let 

y = sin x, and y } = sin (x + h) ; 
.-. y 1 - y = sin (a? + h) sin a?. 
By the formula (36), page 500, 

n h / h\ 

y l - y = 2 sin - cos ( x + ~ ). 

2 \ 2/ 

Divide by h and 

2^L_ JL_ COS ( X + 

h J V 

But the limit of sin 0/0 is unity (page 499), 
T- , (incr. i/} 

Lt h = n -. ?-( = GOSX' 

(incr. x) 

dy d(sin x} 
.'. -= = i-= ' = cos a; (1) 

(2) The differential coefficient of cos x is - sin x. Let 

y = cos x, and y l = cos (a? -f h) ; 

y l - y = COS (a? + /&) - COS X. 

From page 499 y l - y = - 2 sin - sin (x + V 



or 



- a 

2 V. v* 2 

sin h . / h 



7 17 *-*** i i"-' | 

/^ |/z, V 2 

and at the limit when /i = 0, 



(incr. x) 



^% = d(cos*) = _ sina , ^ (2 

da; da; 

The meaning of the negative sign can readily be deduced from 
the definition of the differential coefficient. d(cos x)/dx represents 
the rate at which cos a? increases when x is slightly increased. 
The negative sign shows that this rate of increase is negative, in 
other words, cos a; diminishes as x increases. 

(3) The differential coefficient of tanx is sec 2 x. Using the re- 
sults already deduced for sin x and cos x, 



Gosx 

( d(smx] d(cosx}} f 
J cos OJ-i-g ' - sin x--= ' V / cos 2 a:, 
\ dx dx // 

= (cos 2 x + sin 2 a?)/cos 2 a?. 



15. THE DIFFERENTIAL CALCULUS. 33 

But the numerator is equal to unity (formula (17), page 499). Hence 

d(t&n x) 1 2<r t*\ 

^7 = ^ 2,r. = * ' ' a > 

GW; COS .T 

In the same way d(cot x)(dx = - cosec 2 #. ... (4) 

The remaining trignometrical functions may be left for the 

reader to work out himself. The results are given on page 158. 

EXAMPLES. (1) If y = cos n x, dy/dx - ncos*- 1 ^ . smx. 

(2) If y = sin'x, dy/dx = n sin' - l x . cos x. 

(3) If a particle vibrates according to the equation y = a sin (qt - f), what 
is its velocity at any instant when a, q and e are constant ? 

Ansr. v = dy/dt aq cos (qt - e). 

(4) If y = sin 2 (na - - a), dy/dx = 2wsin (nx - a)cos(nx - a). 

15. The Differentiation of Inverse Trignometrical Functions. 
The Differentiation of Angles. 

The equation, sin# = y, means that x is an angle whose sine is 
y. It is sometimes convenient to write this another way, viz., 

meaning that sin " l y is an angle whose sine is y. Thus if sin 30 = |> 
we say that 30 or sin~H is an angle whose sine is J. Trigno- 
metrical ratios written in this reversed way are called inverse 
trignometrical functions. The superscript " - 1 " has no other 
signification when attached to the trignometrical ratios. Note, if 
tan 45 = 1, then tan ~ l l = 45 ; .-. tan (tan - *1) = tan 45. 

Their differentiation may be illustrated by proving that the 
differential coefficient of sin ~ l x is I/ ,/(! - x 2 ). If y = sin ~ l x t . 
then sin y = x, and 

dx/dy = cos y ; or dy/dx = I/cos y. 
But we know that 

cos' 2 ?/ + sin 2 2/ = 1, or cos y = J(l - sin 2 ?/) = v /(l - # 2 ) v 
for by hypothesis sin y = x. Hence 

dx dx cosy ~ *Jl - x 2 ' 

The ambiguity of the sign means that if any assigned value of 
x satisfies the equation y = sin " l x, so does TT - y, %TT - y and in 
general HIT y. When y has its least value the angle whose sine 
is x is acute. The differential coefficient is then positive, that is 
to say, 

d(Bm-*x) 1 m 

dx ln - "^ 

C 



34 HIGHER MATHEMATICS. 16. 

Similarly d(cos " l x)/dx = - I/ ^/i _ x 2 . - (2) 

The differential coefficient of tan ~ l x is an important function, 

since it appears very frequently in practical formulae. 

If y = tan" 1 ;*?, x = tan?/, dx/dy = 1/cos 2 ^. But (page 499) 

cos 2 # = 1/(1 + tan 2 ?/) = 1/(1 + # 2 ). Hence 

e.^.., . . (3) 



d(cot- l x)/dx = - 1/(1 + # 2 ) . . . (4) 
The remaining inverse trignometrical functions may be left to the 
reader. Their values will be found on page 158. 

EXAMPLES. (1) Differentiate y = sin- ^/^(l+a- 2 ). Put sin y = x\ N /lT^ 
hence cos ?/% - da;/ ^(1 + 2 ) s . But cos y = N /(l - sin 2 ?/) = x /[l - a- 2 /(l + a; 2 )]. 
Substituting this value of cosy in the former result we get, on reduction, 
dyjdx = 1/(1 + a- 2 ) - the answer required. 



<3) If y = tan - 1 ^7==' 3! = x /j-3^r 2 - See formula (19), page 499. 

(4) Ity = tan - *,T + tan - >1, ^ = o. 
x dx 

^5) If y = sin - ^cos x), dy/dx = - 1. 



16. Logarithms and their Differentiation. 

It is proved, in elementary algebra that all numbers may be 
represented as different powers of one fundamental number. E.g., 
1 = 10, 2 = 10- 301 , 3 = 10- 477 , 4 = 10- 602 , 5 = 10' 699 , . . . 

The power, index or exponent is called a logarithm, the funda- 
mental number is called the base of the system of logarithms. 

Thus if 

a' = b, 
x is the logarithm of the number b to the base a, and is written 

x = Iog a 6. 

For convenience in numerical calculations tables are used in 
which all numbers are represented as different powers of 10. 
The logarithm of any number taken from the table thus indicates 
what power of 10 the selected number represents. Thus if 

10 3 = 1000, 10 1 ' 0413927 = 11; 
then 3 = Iog 10 1000, 1-0413927 = Iog 10 ll. 

Exactly the same thing is true if the base 10 be replaced by any 
other base. Read 188. 



gl6. THK DIFFERENTIAL CALCULUS 36 

Before finding the relation between the logarithms of a number 
to different bases, we shall proceed to deduce the differential co- 
efficient of a logarithm. A logarithmic function is any expression 
containing logarithmic terms. E.g., y = logx + x' 3 . 

(1) To determine the differential coefficient of log x. Let 

y = logo;, and y^ = log(x + h). 
Then ft -y = log(x + h) - logs. 

but it is known (page 37) that log a - log b = log -, therefore 

b 

(incr. y) _ 1 /x + h\ 
(incr. x}~ h \ x /' 



and T' ==Lth = Q h^ 

The limiting value of this expression cannot be determined in its 
present form by the processes hitherto used, owing to the nature 
of the terms 1/h and h/x. The calculation must therefore be made 
by an indirect process. See 103. 

Let x = u then 



j log( 1 +-}- . ulogil 4- -), 
h 8 \ x) x 8 \ */ 



= - . logl + 

X b \ U 

As h decreases u increases, and the limiting value of u when 
h becomes vanishingly small, is infinity. The problem now is to 



find what is the limiting value of log( 1 + - J when u is infinitely 
great. That is to say, 

| = ^^.log(l + 3". - (2) 

According to the binomial theorem (page 22) 
1\" // 1 u(u - 1) 1 



dividing out the u's in each term we get 




3! 



36 HIGHER MATHEMATICS. 16. 

The limiting value of this expression when u is infinitely great is 
evidently equal to the sum of the infinite series of terms 

1 + 1 + 2! + 3[ + 47 + ' ' ' to infinitv ( 3 ) 
(see page 230). Let the sum of this series of terms be denoted by 
the symbol e. By taking a sufficient number of these terms we 
can approximate as close as ever we please to the value of e. If 
taken to the ninth decimal place 

e = 2-718281828 . . . 

This number, like TT = 3'14159265 . . ., plays an important roleiu 
mathematics. Both magnitudes are incommensurable and can only 
be evaluated in an approximate way (see page 454). 
Eeturning now to (2), it is obvious that 



This formula is true whatever base we adopt for our system of 
logarithms. If we use 10 

Iog 10 e = 0-43429 . . . = (say) M, 
and % = d(log u x) = M ' 

ax ax x 

Since log a & = 1 (page 480) we can put expression (4) in a much 

simpler form by using a system of logarithms to the base e, then 

dy 



dx dx x 

Logarithms to the base e are called natural or Napierian 
logarithms. Logarithms to the base 10 are called Briggsian 
or common logarithms. 

(2) To find the relation betiveen the logarithms of a number to 
different bases. Let n be a number such that 

a" = n, or a = log a w, 

and (3 b = n, or b = log^w. 

Hence a a = fi b . 

" Taking logs " to the base a 

a = b log a /2 
Substitute for a and b and 

log a w = log^w . log a # or log^w = log a w/log a ^ . (7) 

In words, the logarithm of a number to the base j3 may be obtained 
from the logarithm of that number to the base a by multiplying it 

by j ~ For example, suppose a = 10 and (3 = e, 
logn = Iog 10 ra/log 10 e, 



16. THE DIFFERENTIAL CALCULUS. 37 

where the subscript in log r ?i is omitted. This is the usual practice. 
Hence : 

To pass from natural to common logarithms 

common log = natural log x 0*4343 } ,Y 

Iog 10 a = log,/* x 04343 / ' 

To pass from common to natural logarithms 

natural log = common log x 2-30261 /q, 

log e a = Iog 10 a x 2-3026 J 

The number 04343 is called the modulus of the Briggsian or 
common system of logarithms. When -required it is written M 
or p.* 

EXAMPLES. (1) If y = log aor*, show that dyjdx = 4/a?. 

(2) If y = x n log x, show that dy/dx - x n - l (l + n log .r). 

(3) What is meant by the expression, 2-71828" x 2 - 3W8 = 10"? Ansr. If 
n is a common logarithm, then n x 2-3026 is a natural logarithm. Note, 
e = 2-71828. 

In seeking the differential coefficient of a complex function 
containing products and powers of polynomials, the work is often 
facilitated by taking the logarithm of each member separately 
before differentiation. The compound process is called logarithmic 
differentiation. 

EXAMPLES. (1) Differentiate y x n /(l + x) n . 

Here log y = n log x - n log (1 + .r), or dy\y = ndxlx(\ + .r). Hence 
dy\dx = ynjx(l + x) = nx n - l /(l + x)v + l . 

(2) Differentiate x*(I + x) n /(x s - I). 

Ansr. {(n + I)* 4 - a? - (n + 4)ar - 4}a- 3 .(l + x) - l l(x* - I) 2 . 

(3) Establish (12), 12, by log differentiation. In the same way, show that 

d(xyz) = yzdx + zxdy + xydz. 

The differential coefficient of complex transcendental functions 
can often be easily obtained in this way. 

EXAMPLES. The following standard results can now be verified: 

(1) If y = log sin x, dy/dx = d(sin x)/sin x = cot x ... (10) 

(2) If y = log tan x, dyjdx - 2/sin 2a- . . . . . . (11) 

* Note : the logarithm of the product ab is log a + log b. 

The logarithm of the fraction a/b is log< - log b. 

The logarithm of a poicer, say a n , is n log a, and so on (see page 479). The use of 
logarithms is explained in the introductory pages of the table books. Chambers's 
Afatliematical Tables is a convenient set to have at hand. Less cumbersome and 
cheaper tables are, however, quite as useful for most scientific calculations. See pages 
484 and 520. 



38 HIGHER MATHEMATICS. $ 17. 

(3) If y = log cos x, dy/dx = - tan x (12) 

(4) If y = log cot x, dy/dx = - 2/sin 2x (13) 

(5) Ity = log sin l x, dy/dx = - x/(l - a: 2 ) (14) 

(6) Ity = log cos - 1 ar, dyjdx = xf(l - x 2 ) (15) 

(7) Ity = log tan - 1 a?, dyjdx ='- 2xl(l + x 2 ) (16) 

(8) If y = log cot - *x, <fy/<te = 2<r/(l + z 2 ) (17) 

(9) If y = x(a 2 + x z ) \/a 2 - x 2 , dy\dx = (a* + a 2 x 2 - 4a- 4 )( 2 - x 2 ) - J. 



17. The Differential Coefficient of Exponential Functions. 

Exponential functions are those in which the variable quantity 
occurs in the index. Thus, a x , e x and (a + x) x are exponential 
functions. A few words on the transformation of logarithmic 
into exponential functions may be needed. 

It is required to transform logy = ax into an exponential 
function. Kemembering that log a to the base a is unity, it 
makes no difference to any magnitude if we multiply it by 
such expressions as log a a, Iog 10 10, log e e, and so on. Thus since 
log e (e ax ) = ax log e e, if log e y = ax, 

log e y = ax log e e = log e e ax ; .-. y = e ax , 

when the logarithms are removed. In future " log e " will generally 
be written " log ". 

EXAMPLES. (1) Show that if log y - log y = kct, y = y e ~ kct . 

(2) If log I = - an, I e~ an . 

(3) If 6 = be ~ at , log b - log = at. 

(4) If log eT - = 08, Iog 101 - - -43430. 

J. JU J- *C- 

The differentiation of exponential functions may be conveniently 
studied in three sections : 

(i) Let y = e x . 

Taking logarithms and then differentiating 

dy dy 

logy = x loge; = dx, or ^ = e x 



dx ' 
(ii) Let y = a x . 

As before log y = x log a ; - = y log a ; 



(2) 



18. THE DIFFERENTIAL CALCULUS. 39 

(iii) Let y = x*, 

where x and z are both variable. Taking logarithms and differ- 

entiating 

dy zdx. 

logy = z log x; = log xdz + , 

. . dy = tf log xdz + zx*- l dx ... (3) 
If x and z are functions of t 

dy dz dx 

" 



EXAMPLES. (1) If y = a"*, dyfdx = no, log a. 

(2) If y = (a* + x) 2 , dy/dx = 2(a* + x) (a* log a + 1). 

(3) If y = x*, dy/dx = x*(l - log x}^. 

(4) If y = e<?, dyjdx = e* . e< 

(5) If y = x* t dyjdx = x* . ar*{(log.r) 2 + logz + Ijx}. 

(6) Magnus' empirical formula for the relation between the pressure of 
aqueous vapour and temperature is 

p = aW** 

where a, 6, y are constants. Show that dpjde = 7 g ' fl ? a . by + e . This 

formula represents the increase of pressure corresponding to a small rise of 
temperature from (say) 6 to (6 + 1). 

(7) Blot's empirical formula for the relation between the pressure of 
aqueous vapour (p) and the temperature (6) is 

log p = a + ba.9 - cftO ; hence = pba.0 log o - pc& log j8. 



(8) Required the velocity of a point which moves according to the equation 
y = ae - A* cos 2*Y ^ + Y Since velocity = dyjdt 







^ j 



0082* + f + - sin 



18. The " Compound Interest Law " in Nature. 

I cannot pass by the function e* without indicating its great 
significance in physical processes. From the above equations it 
follows that if 

y = Ce" x ; then dy/dx = be ax ... (1) 
where a, b and C are constants, b, by the way, being equal to 
aC log,e. C is the value of y when x = 0. (Why ?) It will be 
proved later on that this may be reversed, and if 

dy 

f x = 6 then y = CeT*, . (2) 

where a, 6 and C are again constant. 



40 HIGHER MATHEMATICS. 18. 

All these results indicate that the rate of increase of the ex- 
ponential function e x is e x itself. If, therefore, in any physical 
investigation we find some function, say <, varying at a rate 
proportional to itself (with or without some constant term) we guess 
at once that we are dealing with an exponential function. Thus if 

dd> 

jj- = a<f>; we may write < = Ce r<x , or Ce ~ ax , 

according as the function is increasing or decreasing in magnitude. 

Money lent at compound interest increases in this way, and 
hence the above property has been happily styled by Lord Kelvin * 
" the compound interest law ". A great many natural phenomena 
possess this property. The following will repay study : 

ILLUSTRATION 1. Compound interest. If 100 is lent out at 
5 / per annum, at the end of the first year 105 remains. If 
this be the principal for a second year, the interest during that 
time will be charged not only on the original 100, but also on the 
additional 5. To put this in more general terms, let p be lent 
at r / per annum, at the end of the first year the interest amounts 

T 

to ^oT an( ^ ^ Pi be the principal for the second year, we have at 



the end of the first year 

Pi = 
and at the end of the second year, 

p 2 = Pl (l + r/100) = p (l + r/100) 2 . 

If this be continued year after year, the interest charged on the 
increasing capital becomes greater and greater until at the end of 
t years 



Instead of adding the interest to the capital every twelve 
months, we could do this monthly, weekly, daily, hourly, and so 
on. If we are to compare this process with natural phenomena, 
we must imagine the interest is added to the principal continuously 
from moment to moment. Natura non facit saltus. In this way 
we should approximate closely to what actually occurs in Nature. 

As a first approximation, suppose the interest to be added to 
the principal every month. It can be shown in the same way that 
the principal at the end of twelve months, is 

p = Po (l + r/12 . 100) 12 . (4) 

* Quoted from Perry's Calculus for Engineers (E. Arnold, Loiidon). 



18. THE DIFFERENTIAL CALCULUS. 41 

If we consider now the interest is added to the principal every 
moment, say t, we may replace 12 by t, in (4), and 

r \ 
+ lOOTj} 



For convenience in subsequent calculation, let us put JQQ^ = , so 
that t = ur/lQQ. From (5) and formula (16), page 483, 



But (1 + l/u) n has been shown in (3), page 36, to be equivalent 
to e when u is infinitely great ; hence, writing r/100 = x, 

p = p Q e* I ( 6 ) 

which shows that the exponential function represents the rate of 
increase of the principal with time, when the principal is reckoned 
from moment to moment. 

We could deduce this result in a simpler, but perhaps less in- 
structive way. Note that log(l + r/100), and also Iog_p , are 
-constant. Put the former equal to a. From (3) 

dp 

dt = a l } ' 

We guess at once that we are dealing with an exponential func- 
tion. Hence we may put, as on page 40, 

p = Ce at . 
To find the value of C, note that when t = 0, p = p^, and therefore 

P = W"> 
which is identical with (6), when we put x = at. 

ILLUSTRATION 2. Newton's law of cooling. Let a body have a 
uniform temperature O v higher than its surroundings, it is required 
to find the rate at which the body cools. Let denote the tem- 
perature of the medium surrounding the body. 

In consequence of the exchange of heat, the temperature of the 
body gradually falls from O l to Q . Let t denote the time required 
by the body to fall from O l to 0. The temperature of the body is 
then - above that of its surroundings. The most probable 
supposition that we can now make is that the rate at which the 
body loses heat ( - dQ) is proportional to the difference between 
its temperature and that of its surroundings. Hence 



where k is a coefficient depending on the nature of the substance. 



42 HIGHER MATHEMATICS. 18. 

From the definition of specific heat, if s denotes the specific 
heat of unit mass of substance 

Q = 8(6- ), 

or dQ = sdO. ' 

Substitute this in the former expression. Since kfs = constant = 
a (say) and = C. , we obtain, . 

-as-"* ..... < 7 > 

or, in words, the velocity of cooling of a body is proportional to 
the difference between its temperature and that of its surroundings. 
This is Newton's w r ell-known law of cooling (Preston's Theory of 
Heat, p. 444). 

Since the rate of diminution of is proportional to itself, we 
guess at once that we are dealing with the compound interest law,. 
and from a comparison with (1) and (2) above, we get 

B = be ~ * .... (8) 
or log b - log (9 = at. . . . (9) 

If O l represents the temperature at the time t lt and # 2 the 
temperature at the time t z , we have 

log b - log 0j = aj, and log b - log 2 = at. 2 . 
By subtraction 

^7-^.log^, . . . (10) 

h ~ h *i 
a being constant. 

The validity of the original " simplifying assumption''' as to the 
rate at which heat is lost by the body must be tested by comparing 
the result expressed in equation (10) with the results of experiment. 
If the logical consequence of the assumption agrees with facts,. 
there is every reason to suppose that the working hypothesis is 
true. For the purpose of comparison we may use Winkelmann's^/ 
data, published in Wiedemann's Annalen for 1891, for the rate of 
cooling of a body from a temperature of 19'9 C. to 0C.* 

Hence if 6 denote the temperature of the body at any time 
t 2 - t lf anS 2 = 19'9, 6> } = 0, remembering that in practical work 
Briggsian logarithms are used, we obtain, from (10), the expression. 
1 9 

~^ ' log i~ ^ constant > sav k- 



* I was led to select this happy illustration of Newton's law from Winkelmann's- 
papers (Wied. Ann., 44, 177, 429, 1891) after reading Nernst and Schonflies' Intro- 
duction to the Mathematical Treatment of Science. 



is. 



Till: DIFFERENTIAL CALCULUS. 



4:; 



The data are to be arranged as shown in the following table 
(after Winkelmann) : 



0. 


f 2 - ,. 


k. 


18-9 


3-45 


0-006490 


16-9 


10-85 


0-006540 


14-9 


19-30 


0-006509 


12-9 


28-80 


0-006537 


10-9 


40-10 


0-006519 


8-9 


53-75 


0-006502 


6-9 


70-95 


0-006483 



k is therefore constant within the limits of certain small irregular 
variations due to experimental error. Thus the truth of Newton's 
supposition is established. 

This is a typical example of the way in which the logical 
deductions of an hypothesis are tested. 

This can be done another way. Dulong and Petit (Ann. de 
Chim. et de Phys., [2], 7, 225, 337, 1817) have made the series of 
exact measurements shown in columns 1 and 2 of the following 
table : 



t-, velocity of cooling = d6/dt. 


temp, of 
body above 


Calculated by the formula of : 


that of 
medium. 


Observed. 
Newton. 


Dulong and 
Petit 


Stefan. 


220 


8-81 6-88 


8-89 


8-92 


200 7-40 6-20 


7-34 


7-42 


180 6-10 5-58 6-03 


6-09 


160 4-89 4-96 4 -87 


4-93 


140 3-88 4-34 


3-89 3-92 


120 3-02 3-72 3*05 


3-05 


100 


2-30 3-10 


2-33 


2-30 



If we knew the numerical value of the constant a in formula 
(7), this formula could be employed to calculate the value of d6/dt 
for any given value of 0. To evaluate a, substitute the observed 
values of v and in (7) and take the mean of the different results 
so obtained (see 106). Thus, a = 0'31. Column 3 shows the 
velocities of cooling calculated on the assumption that Newton's 
law is true. The agreement between the experimental and theo- 



44 HIGHER MATHEMATICS. 18. 

retical results is very poor. Hence it is necessary to seek a second 
approximation to the true law. 

With this object, Dulong and Pejbit have proposed 

v = b(c e - 1), 

as a second approximation. Here b = 2*037, c = 1-0077. Column 
4 shows the velocity of cooling calculated from Dulong and Pe tit's 
law. The agreement between theory and fact is now very close. 
This formula, however, has no theoretical basis. It is the result 
of a guess. 

Stefan's guess is that 

v = a{(273 + 0) 4 - (273) 4 }, 

where a = 10 ~ 9 x 16-72. The calculated results (column 5) are 
quite as good as those attending the use of Dulong and Petit's 
formula. Galitzine has pointed out that Stefan's formula can be 
established on theoretical grounds. 

It is a very common thing to find different formulae agree, so 
far as we can test them, equally'well with facts The reader must, 
therefore, guard against implicit faith in this criterion the agree- 
ment betiveen observed and calculated results as an infallible 
experimentum crucis. 

A little consideration will show that it is quite legitimate to 
deduce the numerical values of the above constants from the 
experiments themselves. For example, we might have taken the 
mean of the values of k in Winkelmann's table above, and applied 
the test by comparing the calculated with the observed values of 
either t 2 - t lt or of 0. 

EXAMPLE. To again quote from Winkelmann's paper, if, when the tem- 
perature of the surrounding medium is 99 - 74, the body cools so that when 
= 119-97, 117-97, 115-97, 113-97, 111-97, 109-97 ; 
t = 0, 12-6, 26-7, 42-9, 61-2, 83-1. 

Do you think that Newton's law is confirmed by these measurements ? 
Hint. Instead of assuming that = 0, it will be found necessary to retain 
in the above discussion. Do this and show that the above results must be 
tested by means of the formula 

1 . Iog 10 02 ~ 0(> = constant. 
*,-*, ' 10 i-o 

To return to the compound interest law. 

ILLUSTRATION 3. The variation of atmospheric pressure with 
altitude above sea level can be shown to follow the compound 
interest law. Let p Q be the pressure in centimetres of mercury at 



18. THE DIFFERENTIAL CALCULUS. 45 

the so-called datum line, or sea level, p the pressure at a height h 
above this level. Let p be the density of air at sea level (Hy = 1). 
Now the pressure at the se.a level is produced by the weight of 
the superincumbent air, that is, by the weight of a column of air 
of a height h and constant density p . This weight is equal to hp . 
If the downward pressure of the air were constant, the barometric 
pressure would be lowered p centimetres for every centimetre rise 
above sea level. But by Boyle's law the decrease in the density 
of air is proportional to the pressure, and if p denote the density 
of air at a height dh above sea level, the pressure dp is given by 
the expression 

dp = - pdh. 

If we consider the air arranged in very thin strata, we may regard 
the density of the air in each strata as constant. By Boyle's law 

PPo = PoP, or p = Po p/p . 
Substituting this value of p in the above formula, we get 

dp p p n n 

as" "^ 

The negative sign indicates that the pressure decreases vertically 
upwards. This equation is the compound interest law in another 
guise. The variation in the pressure, as we ascend or descend, is 
proportional to the pressure itself. Since p Q /po is constant, we 
have on applying the compound interest law to (11), 

-** 

p = constant x e P( > . 

We can readily find the value of the constant by noting that at 
sea level h = 0, and p = p Q . Substituting these values in the last 
equation, and remembering that e = 1, constant = _p , 

Pr 

a relation known as Halley's law. Continued p. 213. 

ILLUSTRATION 4. The absorption of actinic energy from light 
passing through an absorbing medium. A beam of light of in- 
tensity I is changed by an amount dl after it has passed through 
a layer of absorbing medium dn thick. That is to say 

dl = - aldn, 

where a is a constant depending on the nature of the absorbing 
medium and on the wave length of light. The rate of variation 
in the intensity of the light is therefore proportional to the in- 



46 HIGHER MATHEMATICS. 18. 

tensity of the light itself, in other words, the compound interest 
law again appears. Hence 

-j- = - al ; or I = constant x e an . 
an 

If J denote the intensity of the incident light, then when 

n 0, / = J = constant. 

Hence the intensity of the light after it has passed through a 
medium of thickness n, is 

I = I e~ . (13). 

The student might profitably read Bunsen and Eoscoe's work 
on the absorption of light by different media, in the Philosophical 
Transactions of the Eoyal Society for 1857. 

ILLUSTRATION 5. Wilhelmy' s law for the velocity of chemical 
reactions. Wilhelmy as early as 1850 published the law of mass 
action in a form which will be recognised as still another example 
of the ubiquitous law of compound interest. The statement of 
the law of mass action put forward by Harcourt and Esson is 
probably the simplest possible. It is " the amount of chemical 
change in a given time is directly proportional to the quantity of 
reacting substance present in the system ". (Wilhelmy, Annalen 
der Physik und Chemie, 81, 413, 499, 1850. See page, 197.) 

If x denote the quantity of changing substance, and dx the 
amount of substance which disappears in the time dt, the velocity 
of the chemical reaction is 

dx 

7t = - kx ' 

where k is a constant depending on the nature of the reacting 
substance. It has been called the coefficient of the velocity of the 
reaction. This equation is probably the simplest we have yet 
studied. It follows directly, since the rate of increase of x is 
proportional to x, that 

x = be ~ , 

where b is some constant to be determined.* The negative sign 

indicates that the velocity of the action diminishes as time goes on. 

Harcourt and Esson's papers in the Philosophical Transactions 

for 1866, 1867 and 1895 might be read with advantage and profit. 

EXAMPLES. (1) If a volume v of mercury be heated to any temperature 8, 
the change of volume dv corresponding to a small increment of temperature 
dO, is found to be proportional to v, hence 

dv = avdd. 

* How ? See page 162. 



<].. THK DIFFERENTIAL OALCULU& 47 

Prove /;i.s-.sf7i//'.s fonn/ilti, r = e*9, for the volume of mercury at any tempera- 
ture 6- Ansr. ? = be<*9, where /?, b, a are constants. If 6 = 1 we have the 
iv. | u i red result. 

('2) According to Nardenskjuld's solubility lair, in the absence of super- 
saturation, for a small change in the temperature (dd), there is a change in 
the solubility of a salt (ds) proportional to the amount of salt s contained in 
the solution at the temperature 6, or 

ds tiKtld 

where a is a constant. Show that the equation connecting the amount of 
salt dissolved by the solution with the temperature is, s bca-0, where b is a 
constant. 

(3) If any dielectric (condenser) be subject to a difference of potential, the 
density p of the charge constantly diminishes according to the relation 

p = be ~ at , 

where b is an empirical constant and a is a constant equal to the product 4ir 
into the coefficient of conductivity (c) of the dielectric and the time (t), divided 
by the specific inductive capacity (/*), i.e., b = ktrctlfj.. Hence show that the 
gradual discharge of a condenser follows the compound interest law. Ansr. 
Show dpjdt = p x a negative constant. 

(4) One form of Dalton's empirical law for the pressure of saturated 
vapour (p) between certain limits of temperature (6) is, 

p - aeQ. 
Show that this is an example of the compound interest law. 

(5) The relation between the velocity v of a chemical reaction and tem- 
perature, 0, is LfOL^^f 

log v = a + be, 

where a and b are constants. Show that we are dealing with the Compound 
Interest Law. What is the logical consequence of this law with reference to 
reactions which (like hydrogen and oxygen) take place at high temperatures 
(say 500), but, so far as we can tell, not at ordinary temperatures ? Look up 
*' False Equilibrium" in your Textbook of Physical Chemistry. 

19. Successive Differentiation. 

The differential coefficient derived from any function of a 
variable may be either another function of the variable, or a con- 
stant (page 17). The new function may be differentiated again 
in order to obtain the second differential coefficient. In the same 
way we may obtain the third and higher derivatives. 

EXAMPLE. Let y = x 3 ; 

the first derivative is, d / =3* 2 ; 
dx 

the second derivative is, -^M = 6x ; 
ax a 

the third derivative is, , Jj( = 6 ; 
oa-' 

the fourth derivative is, = 0. 



48 HIGHEE MATHEMATICS. 19. 

It will be observed that each differentiation reduces the index 
of the power by unity. If n is a positive integer tne number of 
derivatives is finite. 

In the symbols (2/)' (y) > tne superscripts simply de- 



note that the differentiation has been repeated 2, 3 ... times. 
In differential notation we may write these results 

d 2 y = 6x . dx 2 * ; d*y = 6dx* * ; . . . 
The successive differential coefficients of 

y = sinx 
are 

y l = cos x ; y 11 = - sin x ; y m = - cos x\ y iv = sin x ; . . . 
The fourth derivative is thus a repetition of the original function, 
the process of differentiation may thus be continued without end r 
every fourth derivative resembling the original function. 

The simplest case of such a repetition is 

y = e *, 

where y 1 = e x ; y 11 = e x ; y m = e x ; . . . 

The differential coefficients are all equal to each other and to the 

original function. 

EXAMPLES. (1) Show that every fourth derivative in the successive 
differentiation of y = cosx repeats itself. 

(2) If y = log ,r, cPy/dx* = - 6/.r '. 

(3) If y = x n , d 4 yldx 4 = n(n - l)(n - 2)(n - 3)x n ~ 4 . 

(4) Ky = x- 2 , dtyldx? = - 24a: - 5 . 

(5) If y = log (x + 1), dtyjdx* - - (x + 1) ~ s . 

Just as the first derivative of x with respect to t measures a 
velocity, the second differential coefficient of x with respect to t 
measures an acceleration. See page 13. 

EXAMPLES. (1) If a material point (P) f move in a straight line A B 
(Fig. 7) so that its distance (s) from a fixed point O is given by the equation 

s = a sin t, 

where a is constant, show that the acceleration due to the force acting on the 
particle is proportional to its distance from the fixed point. 




FIG. 7. 



* Do not confuse these with dy? = 2x. dx; dx 3 = 3x z . dx ; . . . 

f A tnaterial point is a fiction much used in applied mathematics for purposes 
of calculation, just as the atom is in chemistry. An atom may contain an infinite 
number of "material points " or particles. 



20. THE DIFFERENTIAL CALCULUS. 

The velocity is evidently 

v = ds/dt = a cos t ; 
and the acceleration 

dv d*s 



49 



the negative sign showing that the force (/) is attractive, tending to lessen 
the distance of the moving point from O. 

To get some idea of this motion find a set of corresponding values of /, 
s and v as shown in the following table : 



If t = 





I* 


IT 


fir 


2* . . . 


then v = 


a 





-a 





a ... 


and s = 





a 





a 


... 


and / = 





-a 





a 


... 


and P is at 





B 





A 


Oetc. 



A careful study of these facts will convince the reader that the point is 
oscillating regularly in a straight line, alternately right and left of the point O. 

(2) Another illustration of the second derivative. If a body falls from a 
vertical height according to the law 

s = \g#, 

where g represents the acceleration due to the earth's gravity, show that g is 
equal to the second differential coefficient of s with respect to t. 

(3) If the distance traversed by a moving point in the time t be denoted 
by the equation 

s = at 2 + bt + c 
(where a, b and c are arbitrary constants), show that the acceleration is constant. 

The geometrical signification of the second differential coefficient 
is discussed on pages 133 to 135. 



20. Leibnitz' Theorem. 



To find tlte nth differential coefficient of the product of two functions of x 
in terms of the differential coefficients of each function. 

On page 26 the differential coefficient of the product of two variables was 
shown to be 

dy d(uv) du dv 



where u and v are functions of x. By successive differentiation and analogy 
with the binomial theorem it may be shown that 

d(uv) du dv d n ~ l u dv 

~dx^ = V d^ +n dx' dx^-i + ' ' ' + u d^ ' ' W 
This formula, due to Leibnitz, will be found very convenient in Chapter 
VII "How to Solve Differential Equations". The reader must himself 
prove the formula, as an exercise on 19, by comparing the values of 
&(uv)ldx*, d*(uv)ldx* t . . ., with the developments of (x + ft) 2 , (a: + h) 3 , 
of page 22. 

D 



50 HIGHER MATHEMATICS. 21. 

EXAMPLES. (1) If y = x*.e ax , find the value of d*y/dx*. Substitute x* 
and e nx respectively for v and u in (1). Thus, 

v = x* ; .-. dv/dx = 4a; 3 , d*v/dx* = 12a; 2 , d 3 v/dx 3 = 24x ; 
% = e"* ; .. du/dx = ae ax , d^ujdx 2 = a?e ax , d 3 u[dx* = a?e ax . 
From (1) 

d s y _ d z u dv d?u n(n 1) d*v du n(n l)(n 2) dPv . 
dx* ~ ~dx* n dx'dx? 2! daT 2 ' dx + U ~ 31 ~" ' dx* ' 



36aa; 2 + 24a;). 
If we pretend, for the time being, that the symbols of operation 8 

-5-> (T~) ' (w~) ' ^ n ( 2 )' re P resen ^ the magnitudes of an operation, in an 
algebraic sense, we can write 



instead of (2). The expression a + ^ is supposed to be developed by the 

binomial theorem, page 22, and dvjdx, d^/da; 2 , . . ., substituted in place of 

/ d \ / d \2 

\dx) V% \dx) v ' ' ' '' in the result - Equation (3) would also hold good if 3 

were replaced by any integer, say n. This result is known as the symbolic 
form of Leibnitz' theorem. 

(2) If y = logo-, show that d<V/da; 6 = - 



21. Partial Differentiation. 

Up to the present time we have been principally occupied with 
functions of one independent variable x, such that 

u=f(x); 
but functions of two, three or more variables may occur, say 

u =f(x,y,e, .) 

where the variables x, y, z, . . . are independent of each other. 

Such functions are common. As 
illustrations, it might be pointed 
out that the area of a triangle 
depends on its base and altitude, 
JC" the volume of a rectangular box 
depends on its three dimensions, 
and the volume of a gas depends 
FlG - 8 - on the temperature and pressure . 

(1) To find the differential of a function of two independent 
variables. This can be best done in the following manner, partly 




<_!. THK DIFFERENTIAL CALCULUS, 51 

graphic and partly analytical. In figure 8, the area u of the rect- 
angle ABCD, with the sides x, y, is given by the function 

u = xy. 

Since x and y are independent of each other, the one may be sup- 
posed to vary, while the other remains unchanged. The function, 
therefore, ought to furnish two differential coefficients, the one re- 
sulting from a variation in x, and the other from a variation in y. 

First, let the side x vary while y remains unchanged. The 
area is then a function of x alone, y remains constant. 

.-. (du), = ydx, ... . . (1) 

where (du\ represents the area of the rectangle BB'CC". The 
subscript denoting that y is constant. 

Second, in the same way, suppose the length of the side y 
changes, while x remains constant, then 

(du) x = xdy, .... (2) 
where (du) x represents the area of the rectangle DD'CC'. 

Instead of using the differential form of these variables, we 
may write the differential coefficients 

fdu\ .. (du\ 

\-j- I = y, and [ -y- I = x : 

\dxj, \dy) x 

^u , dtt 

S r.-ijj--! 

where is the symbol of differentiation when all the variables, 

ox 

other than x, are constant. Substituting these values of x and y in 
(1) and (2), we obtain 

(du\= ^dx ; (du) x = ^tiy. 

Lastly, let us allow x and y to vary simultaneously, the total 
increment in the area of the rectangle is evidently represented by 
the figure D'ERBCD. 

(incr. u) = BB'CC" + DD'CC' + CC'C"E 

= ydx + xdy + dx . dy. 

Neglecting infinitely small magnitudes of the second order, we get 
du = ydx + xdy ; . . . . (3) 

dtt , ~ou 
du - dx + 



which is also written in the form 

du - 

the former for preference. 



du du 

du - 



52 HIGHER MATHEMATICS. 21. 

In equations (3) and (4) du is called the total differential of the 

^u 
function ; ^~dx the partial differential of u with respect to x when 

y is constant ; and ^rdy the partial differential of u with respect 

to y when x is constant. Hence the rule : 

The total differential of two (or more) independent variables is 
equal to the sum of their partial differentials. 

The physical meaning of this rule is that the total force acting 
on a body at any instant is the sum of every separate action. 
This is nothing more than the so-called principle of .superposition 
of small impulses.* 

"According to this principle, the total force acting on a particle at any 
moment is the sum of all the infinitely small individual actions to wnich 
the particle is subjected. This, in reality, means nothing more than that the 
total differential represents the total change experienced by the mathematical 
function. For instance, if a gas is exposed to variable conditions of tempera- 
ture and pressure, the total change in volume is the sum of the changes 
which occur at a constant temperature and varying pressure, and at a con- 
stant pressure and varying temperature. The total differential, therefore, is 
equal to the sum of the partial differentials corresponding respectively to a 
changing pressure and to a changing temperature. The mathematical process 
thus corresponds with the actual physical change." (Freely translated from 
Nernst and Schonflies' Einfiihrung in die matliematiscJie Behandlung der 
Naturwissenscliaften, p. 180, 1898.) 

In other words, the total change in u when x and y vary is 
made up of two parts : (1) the change which would occur in u if 
x alone varied, and (2) the change which would occur in u if y 
alone varied. 

Total variation = variation due to x alone + variation due to y alone. 

Equation (4) may be written in a more general manner if we 
put u f(x,y), thus 

*.*>* + *&>.. . . ( 

or du = f l (x)dx + f l (y)dy, 

where the meaning of/ 1 (o?) &ndf l (y) is obvious. 

* Ostwald calls this the "principle of the mutual independence of different pro- 
cesses," or the "principle of the coexistence of different reactions," meaning that if a 
number of forces act upon a material particle, each force produces its own motion 
independently of all the others. (Ostwald, Grundriss der allgemeinen Chemie, Walker's 
translation, p. 297.) 



21. THE DIFFERENTIAL CALCULUS. 53 

EXAMPLES. (1) If u = x* + x*y + //" 



,\du = (3x 2 + 2xy)dx + (a- 2 

(2) If u = x log y ; du = log ?/d.r + a- . dy/i/. 

(3) .If M = cos x . sin y + sin a; . cos y, 

du = (dx + dy) (cos x . cos y - sin x sin y) 
= (dx + dy){coa(x + y)}. 

(4) If u = xi>, du = yx* ~ l dx + a* log xdy. 

(5) The differentiation of a function of three independent variables may 
be left as an exercise to the reader. Neglecting quantities of a higher order, 
if u be the volume of a rectangular parallelepiped having the three dimensions 
x, y, z, independently variable, then 

u = xys, 

and du = dx + ~dy + dz ; .... (6) 



or an infinitely small increment in the volume of the solid is the sum of the 
infinitely small increments resulting when each variable changes independently 
of the others. In differential notation show also that 

du yzdx + xzdy + xydz ..... (7) 

(6) If the relation between the pressure p, and volume v, and tempera- 
ture 6 of a gas is given by the formula pv = R(l + aB), show that the total 
change in volume for a simultaneous change of pressure and temperature is 
dv = aR . defp - R(l + o0) . dp Ip*. 

(7) Clairaut's formula for the attraction of gravitation (g) at different 
latitudes (L) on the earth's surface, and at different altitudes (H) above mean 
tide level, is 

g = 980-6056 - 2-5028 cos 2L - 0-000003H, dynes. 

Discuss the changes in the force of gravitation and in the weight of a sub- 
stance with change of locality. Note, " weight " is nothing more than a 
measure of the force of gravitation. 

(2) To find the differential coefficient of a function of two inter- 
dependent variables. If the meaning of the different terms in 

^u. Du., 
du = ^-dx + ^dy 

is carefully noted, it will be found that the equation is really ex- 
pressed in differential notation, not differential coefficients. In 

~dii 
the partial derivative ^,dx, ^u represents the infinitely small change 

that takes place in u when x is increased by an amount dx, y 

^)u 
being constant ; similarly t)u in ^r~dy stands for the infinitely small 

change which occurs when y is increased by an amount dy, x 
being constant. If du is the total increment in a function of 



54 HIGHER MATHEMATICS. 21. 

the variables when each variable is individually increased by an 
amount (du) y and (du) x , then 

du (du) y + (du) x . 
If the variables x and y are both functions of say t, we have 

y = 




and du = 

We may pass directly from differentials to differential co- 
efficients by dividing through with dt, thus 

du _ ~bf(x, y) dx ^f(x, y) dy 
dt ~tix ' dt ty df 

which is more frequently written 

du ~tiu dx ^u dy > du /du\dx /du\dy 
dt ^x ' dt ^y dt dt \dx ) dt \dy / dt 
When there is likely to be any doubt as to what variables have 
been assumed constant, a subscript is appended to the lower 
corner on the right of the bracket. The brackets in the second 
of equations (8) may be omitted, when there is no chance of 
confusing ^u/^x . . . with differential coefficients. 

The most general form of (8) for any number of variables is 
obtained as follows : If 

u = f(x lt x 2 , . . . x n ), 
where x ly x 2 , . . . are functions of x, then 

du ^u dx l ~bu dx. 2 ^u dx n 

~1 = =N~ ~T~~ ~^~ -N ~T + . . . + A ' ~J (y) 

dx ox l dx otc 2 dx ox n ax 

If, at the same time, x lt x. 2 , . . . are functions of y, 

du ~tiu dx, ^u dx.j ^u dx n 

-j- = s -^ + A ^ J +.--+v^-^' (10) 
dy ox l dy ox 2 dy ox n ay 

Equation (8) leads to some interesting results. 

If y = uv, where u and v are functions of x, then tyftv = u 

and tyfbu = v ; substituting these values in (8), and making the 

necessary changes in the letters,* we get our old formula, page 26, 

dy dv du 

^_ .__ n i [ i, 

dx dx dx 
If u is a function of x, such that u = x, 

*y = *y + *n ^ n 2 > 

dx ^x 1)v ' dx' 

^)y dy 

since dx/dx is unity. Note the distinction between ^ and ^. 

* Note that u, x, y, t of (8) are now to be replaced by y, u, v, and x respectively. 

I 



21. THE 1)11 I I.KKMIAL CALCULUS. 55 

If u is constant, 

dy_ty dv 
dt ~ ^ ' dx 
A formula previously obtained in a different way. 

Many illustrations of functions with properties similar to those 
required in order to satisfy the conditions of equation (8) may 
occur to the reader. The following is typical : 

When rhombic crystals are heated they may have different 
coefficients of expansion in different directions. A cubical portion 
of one of these crystals at one temperature is not necessarily 
cubical at another. Suppose a rectangular parallelepiped is cut 
from such a crystal, with faces parallel to the three axes of 
dilation (see Preston's Theory of Heat, p. 199). The volume of 
the crystal is 

V = xyz, 
where x, y, z are the lengths of the different sides. Hence 

^V/tx = yz ; Wfiy = xz ; W/Dz = xy, 

Substitute in (6) and divide by dO, where dO represents a slight 
rise of temperature, then 

dV dx dy dz 
dO = yz dO + xz dO + xy d& 

where dx;dO, dyfdO, dz/dO represent the coefficients of expansion 
(page 7) along the three directions. 

The coefficient of cubical expansion is obtained by putting 
x = y = z = 1, when 

dx dy dz 
a = dO + dd + 'd~6 ) 

where dx/dO or ^x/DO, etc., represent the linear expansions (A) in 
each direction. For isotropic bodies 

dx/de = dyjdO = dz/d6, and hence a = 3A. 

EXAMPLES. (1) Loschmidt and Obermeyer's formula for the coefficient of 
diffusion of a gas at 6 (absolute) is 



760 

where k is the coefficient of diffusion at C. and p is the pressure of the gas. 
Required the variation in the coefficient of diffusion of the gas corresponding 
to small changes of temperature and pressure. Note k and are constant. 

Put a = fc /7600 ; ^de = apn$ n - ] . de > ^dp = a8*dp. But 



dk = d6 + dp. .-. dk = aO- *(npde + edp). 



56 HIGHER MATHEMATICS. 22. 

(2) Biot and Aragd's formula for tlie index of refraction (u) of a gas or 
vapour at 6 and pressure p is 

* " l " r+~^ ' 760' 

where OQ is the index of refraction at 0, a the coefficient of expansion of the 
gas with temperature. What is the effect of small variations of temperature 
arid pressure on the index of refraction ? Ansr. To cause it to vary by an 

u - I/ dp 
amount du = ' 



22. Euler's Theorem on Homogeneous Functions. 

The following discussion is convenient for reference : 
To show that if u is an homogeneous function * of the nth degree, say u = 
where a + ft = n, then 



By differentiation of the homogeneous function, 
u = axa-yfr + bx* l yP l + ...: 
where o + /3 = o 1 + /8 1 = . . . = n, we obtain 



lyp ; and 



Hence 



The theorem may be extended to include any number of variables (see foot- 
note, page 340). 

*&u ^u ~ (3?t 
EXAMPLES. (1) If u = x*y +~xy* + Sxyz, then ^^ + y ~^ + z ^ ~ ^ u - 

Prove this result by actual differentiation. It of course follows directly from 
Euler's theorem, since the equation is homogeneous and of the third degree. 

(2) If u = - 2 2 x^fr + y^7 = u > since the equation is of the first 

degree and homogeneous.. 

(3) Put Euler's theorem into words. Ansr. In any homogeneous function, 
the sum of the products of each variable with the partial differential coefficients 
of tJie original function with respect to that variable is equal to tlie product of 
the original function with its degree. 



* An homogeneous function is one in which all the terms containing the variables 
have the same degree. Examples : x 2 + bxy + z 2 ; x* + xyz* + y?y + x 2 z 2 are homo- 
geneous functions of the second and fourth degrees respectively. 

t The sign "2" is to be read " the sum of all terms of the same type as . . .," 
or here " the sum of all terms containing x, y and a constant ". The symbol " n " is 
sometimes used in the same way for " the product of all terms of the type ". 



$ 24. THE DIFFERENTIAL CALCULUS. 57 

23. Successive Partial Differentiation. 

\ 

We can get the higher partial derivatives by successive differ- 

entiation, uaing processes analogous to those used on page 47. 
Thus when _ rf + . +af y, 



repeating the differentiation, 

g = 2(1 +r>); ^ = 2(1 + 3^), . (2) 

If we had differentiated ~bu[bx with respect to y, and dw/ty with 
respect to x, we should have obtained two identical results, viz. : 



This rule is general. 

The higher partial derivatives are independent of the order of 
differentiation. By differentiation of *&u[bx with respect to y, 

assuming x to be constant, we get &("5~tyi which is written 



-v-^r ; on the other hand, by the differentiation of <Ht/fy with 

<)% 
respect to aj, assuming i/ to be constant, we obtain ^~- That is 

to sa y * 



This was only proved in (3) for a special case. As soon as the 
reader has got familiar with the idea of differentiation, he will no 
doubt be able to deduce the general proof for himself, although it 
is given in the regular text books. The result stated in (4) is of 
great importance. 

24. Exact Differentials. 

To find the condition that u may be a function of x and y in 
the equation du = Mdx + Ndy, .' ... (5) 

where M and N are functions of x and y. 

We have just seen that if u is a function of x and y 

^)u ^u 
ftt-gfc + jpfe ... (6) 

that is to say, by comparing (5) and (6) 



58 HIGHER MATHEMATICS. 25. 

Differentiating the first with respect to y, and the second with 
respeet to x, we have, from (4) 



In the chapter on differential equations this condition is shown 
to be necessary and sufficient in order that certain equations may 
be solved, or " integrated " as it is called. Equation (7) is therefore 
called the criterion of integrability. An equation that satisfies 
this condition is said to be a complete or an exact differential. 
For examples, see page 290. 

25. Integrating Factors. 

The equation 

Mdx + Ndy = . . . . (8) 

can always be made exact by multiplying through with some function 
of x, called an integrating factor : (M and N are functions of x and y.) 

Since M and N are functions of x and y, (8) may be written 

2-5 ' "> 

or the variation of y with respect to x is as - M is to N; that is 
to say, x is some function of y, say 

f(x, y) = a, 
then from (5), page 52, 



By a transformation of (10), and a comparison of the result with (9), 
we find that 

dy__ D/(a?,y) / VO^SO _ _ M /i l\ 

dx~ SaT" / ~ly ~ N 

Hence %^ . ,M ; and ^ . ^, . . (12) 

where /x is either a function of x and y, or else a constant. Multi- 
plying the original equation by the integrating factor /x, and 
substituting the values of /x3/, ^N so obtained in (12), we obtain 

^te + 5 % = 0, 

^x ty y 

which fulfils the condition of exactness. 

EXAMPLE. Show that the equation ydx - .rdy = becomes exact when 

multiplied by I/?/ 2 . 

dM = 1 . 3N = 1 

-dij if ' 9* " if 

Hence 'dM/'dy = 'dN/^x, the condition required by (7). In the same way show- 
that l/xy and 1/x-- are also integrating factors. 



$-2(\. THI-; I)IFFKI;K.\ 1'iAi. CALCULUS. .v. 

26. Illustrations from Thermodynamics. 

It is proved in the theory of differential equations that the 
number of integrating factors for any equation, Mdx + Ndy = 
is infinite. Integrating factors are very much used in solving 
certain forms of differential equations (q.v.), and in certain 
important equations which arise in thermodynamics. Several 
illustrations of partial derivatives will be found in subsequent 
parts of this work. 

The change of state of every homogeneous liquid, or gaseous 
substance is completely denned by some law connecting the 
pressure (p), volume (v) and temperature (0). This law, called the 
characteristic equation, or the equation of state of the substance, 
has the form 

f(p, v, 0) = 0. 

Any change, therefore, is completely determined when any two of 
these three variables are known. Thus, we may have 

p = M<v, 0); v = f 2 (p, 0) ; or $ = f 3 (p, v). (1) 

Confining our attention to the first, we obtain, by partial differen- 
tiation, 



. 
a do ' r 



where the partial derivative 'bpj'bv represents the coefficient of 
elasticity of the gas, "top/M is nothing but the so-called coefficient 
of increase of pressure with temperature at constant volume. If 
the change takes place at constant pressure, dp = 0, and (2) may 
be written 

<h>\ /ty\ / /ty\ 
Wp " VWJ W* 

The subscript is added to show which factor has been supposed 
constant during the differentiation. Note the change of dv/dO to 
dt?/<)0 at constant pressure. Equations (3) state that the coefficient 
of thermal expansion is equal to the ratio of the coefficient of the 
increase of pressure with temperature at constant volume, and the 
coefficient of elasticity of the gas. 

EXAMPLES. (1) Show that a pressure of 120 atmospheres is required to 
keep unit volume of mercury at constant volume when heated 2 C. (Coefficient 
of expansion of Hg = 0-00018, of compressibility 0-000003.) (Planck.) 

(2) J. Thonisen's formula for the amount of heat Q disengaged when one 
molecule of sulphuric acid (H^SO^ is mixed with n molecules of water (H^O) is 
Q = 17860 n/(l-798 + n) cals. 



60 HIGHER MATHEMATICS. 26. 

Put a = 17860 and b = 1-798, for the sake of brevity. If x of H 9 SO 4 be mixed 
with y of T 2 O, the quantity of heat disengaged by the mixture is x times as 
great as when one molecule of H 2 SO 4 unites with y/x molecules of water. 
Substituting y/x = n in Thomsen's formula 

Q = ayj(bx + y) cals. 

If dx of acid is now mixed with x of H Z SO 4 and y of H 2 0, show that the amount 
of heat liberated is 
?)Q 



' dx cals ' 

In the same way the amount of heat liberated when dy of water is added to a 
similar mixture is 

'd ab 



The student of thermodynamics is not likely to meet with anything more 
difficult than the seven following examples : 

(3) Apply equation (3) to the ordinary gas equation 

pv = Re, ...... (4) 

where -B is a constant, p, v, and 6 have their usual meaning. Ansr. 



What does this mean ? 

(4) Verify the following deductions : Let Q, 6, p, v, represent any four 
variable magnitudes whatever. By partial differentiation, 

^Q, 'dQ J 30 



Equate together the second and last members of (5), and substitute the value 
of dp from (2), in the result. Thus, 



Put dv = 0, and divide by de, 
dQ 



Again, by partial differentiation 



Substitute this value of dd in the last two members of (5), 

dQ 



Put dp = 0, and write the result 



By proceeding in this way, the reader can deduce a great number of 
relations between Q, 6, p, v, quite apart from any physical meaning the letters 
might possess. 

If Q denotes the quantity of heat added to a substance during any small 
changes of state, and p, v, 6, the pressure, volume and absolute temperature 
of the substance, the above formulae are then identical with corresponding 
formulae in thermodynamics. Here, however, the relations have been de- 
duced without any reference to the theory of heat. 



S L'IJ. THE DIFFERENTIAL CALCULUS. 61 

Under these circumstances, (dQj'd6)jl9 represents the quantity of heat 
required for a small rise of temperature at constant volume ; (dQI'd6)t is 
nothing but the specific heat of the substance at constant volume, usually 
written C v ; similarly, (dQfd9) p is the specific lieat at constant pressure, written 
C p ; and (dQl^e and (dQfdp) 9 refer to the two latent heats. 

These results may be applied to any substance for which the relation (4) 
holds good. In this case 



(5) A little ingenuity, and the reader should be able to deduce the so-called 
Beech's Theorem : 



employed by Clement and Desormes for evaluating y. See any text-book on 
physics for experimental detail. 

(6) By the definition of adiabatic and isothermal elasticities (page 92), 

E<t> = - v(dpl'dv)^ t and E ff = - v(dpfdv) 0t respectively. 

The subscripts <f> and indicating, in the former case, that there has been 
neither gain nor loss of heat, in other words that Q has remained constant, 
and in the latter case, that the temperature remained constant during the 
process 'dp/'dv. Verify the following reasoning : 

From the first and last members of (5), when Q is constant, 



From (7), (10) and (3), 

0* - C 
Ee ~~ ~ \Vv 

fdQ\ ffdQ\ C p 
= W)p/('de) v = -C, = 7 ........ (12) 

An important result. 



(7) According to the second law of thermodynamics, " the expression 
is a perfect differential ". It is usually written d<j>, where <f> is called the 
entropy of the substance. From the first two members of (5), therefore, 



is a perfect differential. From (7), page 58, therefore, 
dl VQ\ dC. 



where C, has been written for (dQfdO),, L for (dQfdv)e. 

According to the first laic of thermodynamics, when a quantity of heat dQ 
is added to a substance, part of the heat energy d U is spent in the doing of 
internal work among the molecules of the substance, and part is expended in 
the mechanical work of expansion (p.dv) against atmospheric pressure (see 
page 182). To put this symbolically, 

dQ = dU + pdv ; or dU = dQ - pdv. . . . (15) 

Now d U is a perfect differential. This means that however much energy 
U, the substance absorbs, all will be given back again when the substance 
returns to its original state. In other words, U is a function of the state of 



62 HIGHER MATHEMATICS. 8 2G. 

the substance (see page 295). This state is determined, (2) above, when any 
two of the three variables p, v, 0, are known. 

From the first two members of (5), and the last equation of (14), therefore, 
dU = C v .de + L.dv - pdv = C v .dB + (L - p)dr, . (16) 

is a complete differential. In consequence, as before, 

-(I?) ,17) 

\o9/v 

From (14) and (17), 

1/3QN 

-l^r) (18) 

e\dv Je 

a " law " which has formed the starting point of some of the finest deductions 
in physical chemistry (see page 216). 

(8) Establish Mayer's formula, 

C p - C v = R, . . . . . (19) 
for a perfect gas. 

Hints: (i.) Since pv = RB, (dp^d&) v = R/v; .-. (dQfdO), = RB/v = p. (ii.) 
Evaluate dv as in (2), and substitute the result in the second and third 
members of (5). (iii.) Equate dv to zero. Find 'dvj'dd from the gas equation, 
etc. Thus, 

'VQ\ . fdQ\ fdo\ fdQ\ . fdQ\ /cX?\ R = i^Q\ . etc 

(9) Assuming Neiuton's formula that the square of the velocity of pro- 
pagation (V) of a compression wave (e.g., of sound) in a gas varies directly as 
the adiabatic elasticity of the gas (E$) and inversely as the density (p), or 

V 2 oc Efifp ; show that V- oc yR8. 

Hints : Since the compression wave travels so rapidly, the changes of 
pressure and volume take place without gain or loss of heat. Therefore, 
instead of using Boyle's law, pv constant, we must employ ^T = constant 
(page 212). Hence deduce yp = v . dp/dv = E^. Note that the volume varies 
inversely as the density of the gas. Hence, if 

F 2 oc Ej>lp oc E+v oc ypv oc yRB (20) 

Equations (19) and (20) can be employed to determine the two specific heats 
of any gas in which the velocity of sound is known. Let a be a constant to 
be evaluated from the known values of R, 6, F-, 

.-. C v = R/(l - a], and C,, = aC v . 

Boynton has employed van der Waals' equation in place of Boyle's. Per- 
haps the reader can do this for himself. It will simplify matters to neglect 
terms containing magnitudes of a high order (see Boynton, Physical Review, 
12, 353, 1901). 

(10) If y = e-x + fit + y is to satisfy the equation 



show that a 2 = AfP + B&, where a, /8, 7, are constants. 



CHAPTER II. 
COORDINATE OR ANALYTICAL GEOMETRY. 

" Order and regularity are more readily and clearly recognised when 
exhibited to the eye in a picture than they are when presented 
to the mind in any other manner." DR. WHEWELL. 

27. Cartesian Coordinates. 

THE physical properties of a substance may, in general, be con- 
cisely represented by a geometrical figure. Such a figure furnishes 
an elegant method for studying certain natural changes, because 
the whole history of the process is thus brought vividly before 
the mind. At the same time the numerical relations between 
a series of tabulated numbers can be exhibited in the form of a 
picture and their true meaning seen at a glance. 

Let xOx and yOy' (Fig. 9) be two straight lines at right angles 
to each other, and intersecting at the point 0, so as to divide the 
plane of this paper into four quadrants I, II, III and IV. Let 
P l be any * point in the first quadrant rjOx ; draw P l M l parallel 
to On and P^ parallel to Ox. Then, if the lengths OM l and P^ 
are known, the position of the point P with respect to these lines 
follows directly from the properties of the rectangle NP^M-fl 
(Euclid, i., 34). For example, if OM 1 denotes three units, P l M l 
four units, the position of the point P 1 is found by marking off 
three units along Ox to the right and four units along Oy vertically 
upwards. Then by drawing NP l parallel to Ox, and P^M l parallel 
to Oy, the position of the given point is at P lt since, 

P^ = ON = 4 units ; NP l = OM^ = 3 units. 
x'Ox, yOy' are called coordinate axes. If the angle yOx is a 
ri^ht angle the axes are said to be rectangular. Conditions may 
arise when it is more convenient to make yOx an oblique angle, 
the axes are then said to be oblique. xOx' is called the abscissa 

* It is perhaps needless to remark that what is true of any point /< /, <>/'//. 



64 



HIGHER MATHEMATICS. 



27. 



or x-axis, yOy' the ordinate or y-axis. The point is called 
the origin ; OM l the abscissa of the point P, and P l M l the ordi- 
nate of the same point. In referring the position of a point to a 
pair of coordinate axes, the abscissa is always mentioned first, P 1 
is spoken of as the point whose coordinates are 3 and 4 ; it is 
written "the point P(3, 4)". 

In memory of its inventor, Bene Descartes, this system of 
notation is sometimes styled the system of Cartesian coordinates. 

The usual conventions of trignometry are made with respect 
to the algebraic sign of a point in any of the four quadrants. Any 



FIG. 9. Rectangular Cartesian Coordinates. 

abscissa measured from the origin to the right is positive, to the 
left, negative ; ordinates measured vertically upward are positive, 
and in the opposite direction, negative. For example, if a and 
b be any assigned number of units corresponding respectively to 
the abscissa and ordinate of some given point, then the Car- 
tesian coordinates of the point P l are represented as P^a, b), of 
P 2 as P 2 ( - a, b), of P 3 as P 3 ( - a, - b) and of P 4 as P 4 (a, - b). 
Points falling in quadrants other than the first are not often met 
with in practical work. 

Thus, any point in a plane represents two things, (1) its hori- 
zontal distance along some standard line of reference (#-axis), and 



28. 



COORDINATE OR ANALYTICAL GEOMETRY. 



65 



(2) its vertical distance along some other standard line of reference 
(y-axis). 

When the position of a point is determined by two variable mag- 
nitudes (the coordinates), the point is said to be two dimensional. 

We are always making use of coordinate geometry in a 
rough way. Thus, a book in a library is located by its shelf and 
number ; the position of a town in a map is fixed by its latitude 
and longitude ; etc. 

28. Graphical Representation. 

Consider any straight or curved line OP situate, with refer- 
ence to a pair of rectangular co- 
ordinate axes, as shown in figure 
10. Take any abscissae OM V 
OM 2 ,OM Z , . . . OM, and through 



M M 



M draw the ordi- 




M, M, Mj 



nates P^, P 2 M 2 , . . . PM 
parallel to the y-axis. The ordi- 
nates all have a definite value 
dependent on the slope of the 
line* and on the value of the 
abscissae. If x be any abscissa 0' 
and y any ordinate, x and y are 
connected by some definite law called the equation of the curve. 
It is required to find the equation to the curve OP. In the 
triangle 0PM 

PM = OMt&nPOM, 

or y ictana, ..... (1); 

where a denotes the angle POM. But if OM = PM, 



FIG. 10. 



tan POM = - = 



tan 45. 
The equation of the line OP is, therefore, 

y-*; .... (2) 

and the line is inclined at an angle of 45 to the rr-axis. 

It follows directly that both the abscissa and ordinate of a point 
situate at the origin are zero. A point on the #-axis has a zero 



* Any straight or curved line when referred to its coordinate axes, is called a 
curve". 

E 



66 HIGHER MATHEMATICS. 29. 

ordinate ; a point on the ?/-axis has a zero abscissa. Any line 
parallel to the ic-axis has an equation 

y = b; . . . (3) 

any line parallel to the 7/-axis has an equation * 

x = a, .... (4) 

where a and b denote the distances between the two lines and their 
respective axes. 

29. Practical Illustrations of Graphical Representation. 

Suppose, in an investigation on the relation between the 
pressure (p) and the weight (w) of a gas dissolved by unit 
volume of a solution, we obtained the following successive pairs 
of observations, 

p = i, 2, 4, 8 . . . = x. 
. = i 1, 2, 4 . . . =y. 

IBy setting off on millimetre, coordinate or squared paper 

(Fig. 11) points P^i, l), P 2 (2, 1) 
. . . , and drawing a line to pass 
through all these points, we are 
said to plot the curve. This 
has been done in figure 11. The 
only difference between the lines 
OP of figures 10 and 11 is in their 
2 f e tt slope towards the two axes. 

From equation (1) we can put 
FIG. 11. Solution of Gases in liquids. 

w = p tan a, or tan a = \, 

that is to say, an angle whose tangent is J. This can be found by 
reference to a table of natural tangents. It is 26 33' (approxi- 
mately). 

Putting tan a = m, we may write 

w = mp, .... (5) 

where w is a constant depending on the nature of the gas and 
liquid used in the experiment. 

Equation (5) is the mathematical expression for the solubility 
of a gas obeying Henry's law, viz. : "At constant temperature, 
the weight of a gas dissolved by unit volume of a liquid is propor- 
tional to the pressure". The curve OP is a graphical representation 
of Henry's law. 

To take one more illustration. The solubility of potassium 




$-.*. COORIMXATK OR ANALYTICAL (JK()M K 1 K V. (J7 

chloride (A) in 100 parts of water at temperatures (0) between 
and 100 is approximately as follows : 

= 0, 20, 40, 60, 80, 100 = x, 

A = 28-5, 39-7, 49-8, 59-2, 69-5, 79-5 = y. 

By plotting these numbers, as in the preceding example, we obtain 

a curve PQ (Fig. 12) which, instead of passing through the origin 

at 0, cuts the ?/-axis at the point Q such that 

OQ = 28-5 units = b (say). 

If OP' be drawn from the point parallel to PQ, then the equation 
for this line is obviously, from (5), 

A = w0, 
but since the line under consideration cuts the i/-axis at Q, 

A = mO + b, . . . . (6) 

where b = OQ. In these equations, b, A and are known, the 
value of m is therefore obtained by a simple transposition of (6), 

m = (A - b),'0. 

Substituting the values of b and m in (6), we can find the ap- 
proximate solubility of potassium chloride at any temperature (0) 
between and 100 by the relation 

A = 0-51280 + 28-5. 

The curve QP in figure 12 is a graphical representation of the 
variation in the solubility of 
KCl in water at different 
temperatures. 

Knowing the equation to 
the curve, or even the form 
of the curve alone, the pro- ^ 
.bable solubility of KCl for 
any unobserved temperature Q 
can be deduced, for if the 
solubility had been deter- 
mined every 10 (say) instead 



p 



~20 fd 6ff ~80 TOO 



OA o ,1 -, FIG. 12. Solubility Curve for Klin water. 

of every 20 , the correspond- 

ing ordinates could still be connected in an unbroken line. The 
same relation holds however short the temperature interval. From 
this point of view the solubility curve may be regarded as the path 
of a point moving according to some fixed law. This law is defined 
by the equation to the curve, since the coordinates of every point 
on the curve satisfy the equation. The path described by such a 
point is called the picture, locus or graph of the equation. 



68 HIGHER MATHEMATICS. 30. 

EXAMPLES. (1) Let the reader procure some " squared " paper and plot : 
y = \x - 2 ; 2y + 3x = 12 . . . 

(2) The following experimental results have been obtained : 

When x = 0, 1, 10, 20, 30, ... 

y = - 3, 1-56, 11-40, 25-80, 40-20, . . . 

(a) Plot the curve, (b) Show (i.) that the slope of the curve to the z-axis 
is 1-44 = tan a = tan 60 (nearly), (ii.) that the equation to the curve is 
y = l-44x - 3. (c) Measure off 5 and 15 units along the ic-axis, and show 
that the distance of these points from the curve, measured vertically above 
the a;-axis, represents the corresponding ordinates. (d) Compare the values 
of y so obtained with those deduced by substituting x = 5 and x = 15 in the 
above equation. 

Note the laborious and roundabout nature of process (c) when contrasted 
with (d). The graphic process, called graphic interpolation (q.v.), is seldom 
resorted to when the equation connecting the two variables is available, but 
of this anon. 

(3) Get some solubility determinations from any chemical text-book and 
plot the values of the composition of the solution (C, ordinate) at different 
temperatures (6, abscissa), e.g., Loewel's numbers for sodium sulphate are 

C = 5-0, 19-4, 55-0, 46-7, 44-4, 43-1, 42-2 ; 
= 0, 20, 34, 50, 70, 90, 103-5. 
What does the peculiar bend at 34 mean ? 

In this and analogous cases, a question of this nature has to be decided : 
WJiat is tlw best way to represent the composition of a solution ? Several 
methods are available. The right choice depends entirely on the judgment, 
or rather on the finesse, of the investigator. Most chemists (like Loewel 
above) follow Gay Lussac, and represent the composition of the solution as 
"parts of substance which would dissolve in 100 parts of the solvent". 
Etard found it more convenient to express his results as " parts of substance 
dissolved in 100 parts of saturated solution ". 

The right choice, at this day, seems to be to express the results in mole- 
cular proportions. This allows the solubility constant to be easily compared 
with the other physical constants. In this way, Gay Lussac's method be- 
comes " the ratio of the number of molecules of dissolved substance to the 
number, say 100, molecules of solvent"; Etard's "the ratio of the number 
of molecules of dissolved substance to any number, say 100, molecules of 
solution ". 

30. General Equations of the Straight Line. 

If equations (l)-and (6) be expressed in general terms, using 
x and y for the variables, ra and b for the constants, we can 
deduce the following properties for straight lines referred to a pair 
of coordinate axes. 

(1) A straight line passing through the origin of a pair of 
rectangular coordinate axes, is represented by the equation 

y = mx, . . . . (7) 



30. COORDINATE OR ANALYTICAL GEOMETRY. 

where m = tan a = y/x, a constant representing the slope of the 
curve. The equation is obtained from (5) above. 

(2) A straight line which cuts one of the rectangular coordinate 
axes at a distance b from the origin, is represented by the equation 

y = mx + b . . . . (8) 

where m and b are any constants whatever. For every value of 
m there is an angle such that tan a = m. The position of the 
line is therefore determined by a point and a direction. Equation 
(8) follows immediately from (6). 

(3) A straight line is always represented by an equation of the 
first degree, 

Ax + By + C = ; . . . (9) 

and conversely, any equation of the first degree between two variables 
represents a straight line* 

This conclusion is drawn from the fact that ' any equation 
containing only the first powers of x and 
y, represents a straight line. By sub- 
stituting m = - A IB and b = - C/B in 
(8), and reducing the equation to its 
simplest form, we get the general equa- 
tion of the first degree between two vari- 
ables : Ax + By+C = Q. This represents 
a straight line inclined to the positive 
direction of the #-axis at an angle whose 
tangent is - A/B, and cutting the ^/-axis 
at a point - C/B above the origin. 

(4) A straight line which cuts each coordinate axis at the re- 
spective distances a and b from the origin, is represented by the 
equation 




Consider the straight line AB (Fig. 13) which intercepts the 
x- and i/-axes at the points A and B respectively. Let OA = a, 
OB = b. From equation (9) if 

y = 0, x = a; Aa + C =0,a = - C/A. 
Similarly if x = 0, y = b ; Bb + C = 0, b = - C/B. 

* If the reader has not previously met with the idea conveyed by a "general 
equation," he must pay careful attention to it now. By assigning suitable values to 
the constants . I , /;, ( \ he will be able to deduce every possible equation of the first 
degree between the two variables x and /. See page 481. 



70 HIGHER MATHEMATICS. 30. 

Substituting these values of a and b in (9), i.e., in 
A B x y 

~ C^ ~ C^ = ' we et ~a + ~b = 

There are several proofs of this useful equation. Formula (10) is 
called the intercept form of the equation of the straight line, equa- 
tions (7) and (8) the tangent forms. 

Many equations can be readily transformed into the intercept 
form and their geometrical interpretation seen at a glance. For 
instance, the equation 

x + y = 2 becomes \x + \y = 1, 

which represents a straight line cutting each axis at a distance of 
2 units from the origin. 

One way of stating Gay Lussac' s law is that " the pressure of 
a given mass of gas at constant volume varies directly as the tem- 
perature". If, under these conditions, the temperature be raised 
0, the pressure increases the ^ ? 0rd part of what it was at the 
original temperature.* Let the original pressure, at C., be unity ; 
the final pressure p lt then at 6 

Pi = 1 + ****. 

This equation resembles the intercept form of the equation of a 
straight line (10) where a = 273 and b = I. 
i* The intercepts a and b may be found by 
putting x and y, or rather their equiva- 
lents, 6 and p, successively equal to zero. 
If e = 0, p = I ; if p = 0, = - 273, the 
well-known absolute zero. 

/ a 1 If possible let fall below - 273, then 
FIG. 14 we have a negative value of p in the above 

(b much exaggerated). equation> which is physically impossible. 

The physical signification of this is that temperatures below - 273 
are impossible, if the gas obeys Gay Lussac's law at temperatures 
approaching the absolute zero. 

* Many students, and even some of the textbooks, appear to have hazy notions oil 
this question. According to Gay Lussac' s law, the increase in the volume of a gas at 
any temperature for a rise of temperature of 1, is a constant fraction of its initial 
volume at QC. ; Dalton's law, on the other hand, supposes the increase in the volume 
of a gas at any temperature for a rise of 1, is a constant fraction of its volume at ttiat 
temperature (the " Compound Interest Law," in fact). The former appears to approxi- 
mate closer to the truth than the latter. Gay Lussac says that he got the idea from 
Charles, hence this property of gases is sometimes called Claries' law, or the law of 
Charles and Gay Lussac. 




31. 



COORDINATE OR ANALYTICAL GEOMETRY. 



71 



EXAMPLES. (1) To find the angle between the point of intersection of two 
straight lines whose equations are given. Let the equations be 

y = mx + b ; y' = m'x' + b'. 

Let <f> be the angle required (see Fig. 15), m = tan a, m' = tan a'. From Kuclid, 
i., 32, a' - a = </>, .'. tan (a' - a) = tan 0. By formula, page 500, 

tan a' - tan a m' - m 
tano=, . .._ t ana' = l + mm' ' ' ' ( U > 





FIG. 15. 



N 
FIG. 16. 



(2) To find tlie distance between two points in terms of their coordinates. 
In Fig. 16, let P(x l y l ) and Q(x<fl^ e the given points. Draw QM 1 parallel to 
NM. OM = a-!, PM = Vl ; ON** z 2 , QN = ?/ 2 ; 

WP = MP - MM* = MP - NQ = y, - y 2 ; 

QM 1 = MN = OM - ON - x l - .r 2 . 
Since QPM 1 is a right-angled triangle 



(12) 



31. Differential Coefficient of a Point moving on a Straight 

Line. 

If the amount of gas (v-^ consumed in a burner is proportional 
to the time (^), equal amounts of gas are consumed in equal times, 
Suppose that the amount of gas burnt in one second be denoted 
by V, then for time t Q , v l has a value V Q , and the gas consumed 



in j - t Q seconds amounts to V(t l - 



Hence 



whatever be the values of v and t. This equation can be written 

(i - *o) = V (^ ~ > 
which resembles the equation to a straight line (7), when the 

ratio of the increments of x and y possesses a constant value. 
Expressing the last equation in general symbols, we can put 



_ 

y y increment y 



= constant, 



72 HIGHER MATHEMATICS. 32. 

or, at the limit, the velocity of gas consumption may be represented 

by 

F- 3? -tan; . . . (13) 

that is to say, by a straight line with a slope, or inclination to the 
a;-axis equal to tan a. 

EXAMPLE. Malard and Le Chatelier represent the relation between the 
molecular specific heat (s) of carbon dioxide and temperature (6) by the ex- 
pression 

s = 6-3 + 0-005640 - 0-000001, 080 2 . 

Plot the 6,ds/de-cur\e from 6 = to 6 = 2,000 (abscissae). Possibly a few 
trials will have to be made before the "scale" of each coordinate will be 
properly proportioned to give the most satisfactory graph. The student must 
learn to do this sort of thing for himself. What is the difference in meaning 
between this curve and the s,6 - curve ? 

32. Straight Lines Satisfying Conditions. 

The reader should work through the following examples so as 
to familiarise himself with the conceptions of coordinate geometry. 
Many of the properties here developed for the straight line can 
easily be extended to curved lines. 

(1) The condition that a straight line may pass through a given 
point. This evidently requires that the coordinates of the point 
should satisfy the equation of the line. Let the equation be in the 
tangent form 

y = mx + b. 
If the line is to pass through the point (x lt y-^, 

y l = mx l + b, 

and by subtraction (y - y^} = m(x - x) . . . (14) 

which is an equation of a straight line satisfying the required 
conditions. 

(2) The condition that a straight line may pass through two 
given points. Continuing the preceding discussion, if the line is 
to pass through (x 2 , y 2 ), substitute x 2 , y 2 , in 



' m = (2/2 - 

Substituting this value of m in (14), we get the equation, 



2/2 -2/i x- 2 - i 
for a straight line passing through two given points (x v yj and 

fe> 2/ 2 )- 



8 32. COORDINATE OR ANALYTICAL GEOMETRY. 7:, 

tfp 

(3) To find the coordinates of the point of intersection of two~ >wfe', 
given straight lines. Let the given equations be 
y = mx + b and y = m'x + b'. 

Now each equation is satisfied by an infinite number of pairs of 
values of x and y. These pairs of values are generally different 
in the two equations, but there can be one, and only one pair of 
values of x and y that satisfy the two equations, that is, the 
coordinates of the point of intersection. The coordinates at this 
point must satisfy the two equations, and this is true of no other 
point. 

The roots of these two equations, obtained by a simple algebraic 
operation, are the coordinates of the point required. The point 
whose coordinates are (b' - b)((m - m'), (b'm - bm')((m - m') 
satisfies the two equations. 

(4). To find the condition that three given lines may meet at a 
point. The roots of the equations of two of the lines are the co- 
ordinates of their point of intersection, and in order that this point 
may be on a third line the roots of the equations of two of the 
lines must satisfy the equation of the third. 

EXAMPLE. If three lines are represented by the equations 5.r + 3y = 7, 
3x - y - 10, and x + 2y = 0, show that they will all intersect at a point 
whose coordinates are x = 2 and y = - 1. Solving the last two equations, 
we get x = 2 and y = - 1, but these values of x and y satisfy the first equation, 
hence these three lines meet at the point (2, - 1). 

(5) To find the condition that two lines may be parallel to one 
another. Since the lines are to be parallel they must make equal 
angles with the ic-axis, 

.*. angle a' = angle a, or tana' = tana, 

or m = m, (16) 

that is to say, the coefficient of x in the two equations must be 
equal. 

(6) To find the condition that two lines may be perpendicular to 
one another. If the angle between the lines is 

< = 90 [see (11)] a - a = 90, 
,/. tana' = tan(90 - a) = - cot a = - I/tan a, 






. - . (17) 

or, the slope of the one line to the #-axis must be equal and 
opposite in sign to the reciprocal of the slope of the other. 



HIGHER MATHEMATICS. 



33. 



33. Changing the Coordinate Axes. 

In plotting the graph of any function, the axes of reference 
should be so chosen that the resulting curve is represented in the 
most convenient position. It is frequently necessary to pass from 
one system of coordinate axes to another. In order to do this 
the equation of the given line referred to the new axes must be 
deduced from the corresponding equation referred to the old set 
of axes. 

(1) To transform from any system of coordinate axes to another 
set parallel to the former but having a different origin. Let Ox, 
Oy (Fig. 17) be original axes, and KO^x^ HO^^ the new axes 
parallel to Ox and Oy. Let MM,P be the ordinate of any point P 
parallel to the axes Oy and 0^y r Let h, k be the ordinates of the 



y 


X 




,P 




1 * 


o, 


;M, 


h 


S 



FIG. 17. Transformation 
of Axes. 




FIG. 18. Transformation of Axes. 



new origin O l referred to the old axes. Let (x, y) be the coordinates- 
of P referred to the old axes Ox, Oy, and (x^^ its coordinates 
referred to the new axes. Then OH = h, 0,H = k, 

x = OM = OH + HM = OH + 0^ = h + x l ; 

y = MP = MM, + M,P = 0^ + Mf = k + y r 
That is to say we must substitute 

x = h + x 1 and y = k + y v . . (18) 
in order to refer a curve to a new set of rectangular axes. The 
new coordinates of the point P being 

x l = x - h and y 1 = y - k. . . (19) 
(2) To transform from one set of axes to another having the same 
origin but different directions. Let the t\vo straight lines x,0 and 
y t O, passing through (Fig. 18), be taken as the new system of 
coordinates. Let the coordinates of the point P (x, y) when re- 
ferred to the new axes be x v y v Draw PM perpendicular to 



COORDINATi: <)R ANALYTICAL GEO.MKI UV. 



the old z-axis, and P3f t perpendicular to the new axes, so that 
the angle MPM l = ROM l = a, 

OM = x, OM l = x lt PM = y, PM l = y r 

Draw M V R perpendicular and QM l parallel to the .r-axis. Then 
x = OM = OR - MR = OR - QM V 

= OM l cos a - M^ sin a ; 

.-. x = a^cosa - y l sina .... (20) 
Similarly y = MP = MQ + QP = BM 1 + QP, 

= OM l sin a + M jP cos a, 

.-. y = x l sina + y l cosa .... (21) 
Equations (20) and (21) enable us to refer the coordinates of a 
point P from one set of axes to another. Solving equations (20) 
and (21) simultaneously, 



(22) 



x l = x cos a + y sin al 

y l = y COS a - X sin a J 

(3) To transform from one set of axes to another set having a 
different origin and different directions. Obviously this can be done 
by making the two preceding transformations one after another. 

34. The Circle and its Equation. 

To find the equation of a circle referred to its centre as origin* 
Let r be radius of the circle (Fig. 19) whose centre is the origin of 
the rectangular coordinate axes 
xOx' and yOy'. Take any point 
P(x y y) on the circle. Let PM 
be the ordinate of P. From the 
definition of a circle OP is con- 
stant and equal to r. Then by 
Euclid, i., 47. 

(OM) 2 + (MP) 2 = (OP)' 2 , 
or x 2 + y 2 = f 2 , . (1) 

which is the equation required. 

In connection with this equa- 
tion it must be remembered that 
the abscissae and ordi nates of 
some points have negative 

values, but, since the square of a negative quantity is always 
positive, the rule still holds good. Equation (1) therefore expresses 
the geometrical fact that all points on the circumference are at an 
equal distance from the centre. 




76 



HIGHER MATHEMATICS. 



35. 



EXAMPLES. (1) Required the locus of a point moving in a path according 
to the equations y = a cos t, x = a sin t, where t denotes any given interval of 
time. Square each equation and add, 

y 2 + x 2 = a 2 (cos 2 t + sin 2 ^). 

The expression in brackets is unity (formula (17) page 499), and hence for all 
values of t 

y 2 + x* = a 2 , 
i.e., the point moves on the perimeter of a circle of radius a. 

(2) To find the equation of a circle, whose centre, referred to a pair of 

rectangular axes, has the coordinates h and k. From (19), previous paragraph, 

(x - h) 2 + (y - k) 2 = r 2 , . . . . (2) 

where P(x, y) is any point on the circumference. Note the product xy is 
absent. The coefficients of x 2 and y 2 are equal in magnitude and sign. 
These conditions are fulfilled by every equation to a circle. Such is 
3x 2 + 3y 2 + Ix - 12 = 0. 



The general equation of a circle is 

x 2 + y 2 + ax + by + c = 0*. 



(3) 



35. The Parabola and its Equation. 

There is a set of important curves whose shape can be obtained 
by cutting a cone at different angles. 
Hence the name conic sections. They 
include the parabola, hyperbola and 
ellipse, of which the circle is a special 
case. I shall very briefly describe 
their chief properties. 

A parabola is a curve such that 
any point on the curve is equi-distant 
from a given point and a given straight 
line. 

The given point is called the 
focus, the straight line the directrix, 
the distance of any point on the 
curve from the focus is called the 
FIG. 20. The Parabola. j ocal ra di m . Q, Fig. 20, is called 

vertex of the parabola. AK is the directrix ; OF, PF, P^ . . . 
are focal radii ; 

K Z P S = P Z F, K 2 P 2 = P 2 F, KP = PF, AO = OF. 




The reader should verify all these equations by plotting on his " squared " paper. 



36. COORDINATE OR ANALYTICAL GEOMETRY. 77 

(1) To find the equation of tfie parabola. Take vertex as origin of the 
coordinate axes. Let OA = OF = a. Take any point P(x t y) 

FP = PK = AM = AO + OM= x + a ; 
FM = OM - OF = x - a ; 

PM=y. 
In right-angled triangle FPM 

(x - a) 2 + 7/ = (x + a) 2 ; 

.-. y 2 = lax, (1) 

which is the standard equation of the parabola. The abscissae are proportional 
to the squares of the ordinates. 

(2) To find the shape of the parabola. From (1) 

y = -4-2 *Jax. 

1st. Every positive value of x gives two equal and oppo- 
site values of y, that is to say, there are two points at equal 
distances perpendicular to the #-axis. This being true for all 
values of x, the part of the curve lying on one side of the ic-axis 
is the mirror image of that on the opposite side * ; in this case the 
a;-axis is said to be symmetrical with respect to the parabola. 
Hence any line perpendicular to the #-axis cuts the curve at two 
points equidistant from the ic-axis. 

2nd. When x = 0, the ?/-axis is tangent ) to the curve. 

3rd. a being positive when x is negative, there is no real value 
of y, for no real number is known whose square is negative ; in 
consequence, the parabola lies wholly on the right side of the 2/-axis. 

4th. As x increases without limit, y approaches infinity, that is 
to say, the parabola recedes indefinitely from the x or symmetrical- 
axis on both sides. 

EXAMPLES. (1) By a transformation of coordinates show that the para- 
bola represented by equation (1), may be written in the form 

x = a + by + cy z , . . . . . (2) 

where a, b, c are constants. Let x become x + h; y = y + k\ a = j where fe, k 
and ./ are constants. Substitute the new values of x in (1) and multiply out. 
Collect the constants together and equate to a, b and c as the case might be. 

(2) In the general equation of the second degree 

ax* + bxy + cy z + fx + gy + h = 0, . . . (3) 

if 6 2 - 4oc = 0, the equation represents a parabola, one or two straight lines 
or an impossible curve. Trace the curve x 2 - <xy + y z - 8x + 16 = and 
show that the curve is a parabola, 6 = 2, a = 1, c = 1. What relations must 
exist between the coefficients in order that (3) may represent a circle ? 

* The student of stereo-chemistry would say the two sides were " enautiomorphic ". 
fA "tangent" is a straight line which touches but does not cut the curve (see 
pages 82 and 494). 



78 



HIGHER MATHEMATICS. 



36. 



36. The Ellipse and its Equation. 

An ellipse is a curve such that the sum of the distances of any 
point on the curve from two given points is always the same. 

In Fig. 21 let P be the given point which moves on the curve 
PP' so that its distance from the two fixed points F lt F 2 , called the 
foci, has a constant value say 2a. The distance of P from either 
focus is called the focal radius (or radius vector). 




FIG. 21. The Ellipse. 

(1) To fit id the equation of the ellipse. This is rather a tedious deduction. 
If desired, the final equation may be taken without proof. In the same figure, 
let xOx' and yOy' be a pair of coordinate axes such that the centre bisects 
the line F 2 F l on the x-axis. Take any point P(x, y) on the ellipse. Complete 
Fig. 21. Let OF Z = OF^ = c so that a > c, otherwise F 2 and F 1 fall outside 
the ellipse. Let F^P = r 2 , F^P = r r In the right-angled triangles PF 2 lfand 



= (PIf) 2 + (F 2 M) 2 , and 
or r 2 2 = 3/2 + (x _ c)2) and ri 

Add and also subtract equations (1), when 



ix ; or (r, -f / - 2 ) (i\ - r 2 ) = 4cx. 
By definition of the ellipse r, + r 2 = 2<x and substituting this value of 
in (3), we get 

r l - r z = 2cx/a 

Adding and subtracting r l + r 2 = 2a from this equation we obtain 

T-J = a + cx\a ; r 2 = cx/a -a 

Squaring equations (5), and substituting in (2), we get 

a 4 + c 2 ar 2 = a 2 (7/ 2 + x 2 + c 2 ) ; 
or x 2 (a 2 - c 2 ) + a 2 ^ 2 = 2 (a 2 - c 2 ) 



$36. COORDINATE OR ANALYTICAL GEOMKTKY. 7'' 



in the right-angled triangles P l OF l and P^OF^ 6 2 + c a = 2 , nr <- 
Substituting 6 2 for a 2 - c 2 , in (6), and dividing by a 2 6' 2 , we get 

x 2 ?/ 2 

rf+F- 1 ' ..... (7) 

which is the required equation of the ellipse. 

Obviously, if a = b, this equation passes into that of a circle (page 75). 
The circle is thus a special case of the ellipse. 

The line P 2 P 4 , in Fig. 21, is called the major axis, P^P* the 
minor axis, their respective lengths being 2a and 2ft ; the magni- 
tudes a and b are the semi-axes ; each of the points P l? P.,, P A , P 4 , 
is a verft'.i . 

(2) To % /md the shape of the ellipse. From equation (7) it 
follows that 

y = b x/1 - 2? 2 /a 2 , and # = a v /l - y 2 /6*. . (8) 

1st. Since y 2 must be positive, # 2 /a 2 > 1, that is to say, x 
cannot be numerically greater than a. Similarly it can be shown 
that ?/ cannot be numerically greater than b. 

2nd. Every positive value of x gives two equal and opposite 
values of y, that is to say, there are two points at equal distances 
perpendicularly above and below the iP-axis. The ellipse is there- 
fore symmetrical with respect to the #-axis. In the same way, it 
can be shown that the ellipse is symmetrical with respect to the 
t/-axis. 

3rd. If the value of x increases from zero until x = + a, then 
y = 0, and these two values of x furnish two points on the <r-axis. 
If x now increases until x > a, there is no real corresponding 
value of y 2 . Hence the ellipse lies in a strip bounded by the 
limits x = a ; similarly it can be shown that the ellipse is 
bounded by the limits y = b. 

The ellipse is not a very important curve. Its chief application 
will be discussed later on. 

EXAMPLES. (1) Let the point P(x, y) move on a curve so that the position 
of the point, at any moment, is given by the equations, .r = a cos / and 
y = b sin t ; required the path described by the moving point. 

Square and add, since cos 2 ^ -i- sin 2 ^ is unity (page 499), 



The point therefore moves on an ellipse. 

(2) The general equation of the second degree, 

rt.,-2 + I).,.,/ + cy'2 + f x + gy + /, = Q, 

represents an ellipse when 6 2 - 4ac is negative, or else it represents a circle, 
point, or an imaginary curve. For instance, a >2 - 2a-y + '2y- - .r -- // -r ^ = 
Here 6 2 - ac = - 4. Plot the curve to this equation. 



80 



HIGHER MATHEMATICS. 



37. 



(3) Find the relation between the constants a, b, in, c in the equations 
x z /a z + 7/ 2 /6 2 = 1 and y = mx + c, in order that the line may cut the ellipse in 
two, one, or no point. For the first a 2 w 2 + b 2 - c- must be greater than zero, 
for the second, equal to zero, for the third, less than zero. 

37. The Hyperbola and its Equation. 

The hyperbola is a curve such that the difference of the distance 
of any point on the curve from two fixed points is always the same. 

Let the point P(x, y) (Fig. 22) move so that the difference of 
its distances from two fixed points F, F' (called the foci) is equal 
to 2a. Then PF' - PF = 2a. 

(1) To find the equation of the hyperbola. Let xOx', yOy' be rectangular 
axes intersecting at a point midway between F' and F so that OF = OF' = c r 
and let FP = r, F'P = r'. In the right-angled triangles FPM and F'PM, 

(FP)* = (PM)* + (MF) Z , and (F'P)* = (PM)* + (F^M)* ; 
or r 2 = 7/ 2 + (x - c) 2 , and r'* = y*+(x + c)' 2 . ...(!> 




FIG. 22. The Hyperbola. 

Adding and subtracting equations (1) we get 

r 2 + r' 2 = 2(7/ 2 + x 2 + c 2 ) (2) 

r '2 _ r 2 = CXj or ( r ' _ r ) ( r + r ') = cx . ... (3) 

By definition of the hyperbola, r' - r - 2a. Substituting this result in (3) we 

get 

r + r' = Zcxja (4) 

By addition and subtraction of r' - r - 2a, from (4), 

r' = a + cx/a ; r = - a + cxja. ... (5) 

Squaring equations (5), and substituting in (2), we get 

4 + C 2 a .2 = a 2^2 + X 2 + C 2) . 

or x 2 (a z - c 2 ) + aV = 2 (a 2 - c 2 ). . . . (6) 

By Euclid, i., 20 (Cor.), the difference between any two sides of a triangle is 
smaller than the third side, and therefore 

2a < 2c, or a < c. 



37. COORDINATE OR ANALYTICAL GEOMETRY. 81 

Let * c~ = a 2 + ft 2 , or a* - c 2 = - 6 12 . 

Substituting this value of 6 2 in (6) and dividing out, we obtain the equation 

to the hyperbola in the simple form 

x 2 if- 

a-*- 1 ...... m 

The xOx'-&xis is called the transverse or real axes of the hyper- 
bola ; yOy' the conjugate or imaginary axes ; the points A, A' are 
the vertices of the hyperbolas, a is the real semi-axis, b the 
imaginary semi-axis. 

(2) To find the shape of the hyperbola. From (7) 



+ b 2 . (8) 



_ 
~ v x 2 - a 2 , and x = 




1st. Since y 1 must be positive, x 2 <4 a 2 , or x cannot be nu- 
merically less than a. No limit with respect to y can be inferred 
from equation (8). 

2nd. For every positive value of x, there are two values of y 
differing only in sign. Hence these two points are perpendicular 
above and below the #-axis, that is to say, the hyperbola is sym- 
metrical with respect to the #-axis. There are also two equal and 
opposite values of x for all values of y. The hyperbola is thus 
symmetrical with respect to the ?/-axis. 

3rd. If the value of x changes from zero until x =* + a, then 
y = 0, and these two values of x furnish two points on the re-axis. 
If x > a, there are two equal and opposite values of y. Similarly 
for every value of y there are two equal and opposite values of x. 
The curve is thus symmetrical with respect to both axes, and lies, 
beyond the limits x = a. 

EXAMPLES. (1) In the general equation of the second degree, 

ax 2 + bxy + cy z + fx + gy + h = 0, 

if 6 2 - 4ac is positive, the equation either represents an hyperbola or two 
intersecting straight lines. E.g., x- - 6xy + y 2 + 2x + 2y + 2 = 0. Plot 
this curve. 

(2) The equation to the hyperbola whose origin is at its vertex is 



. 
Substitute x + a for x in the regular equation. Note that y does not change. 

Before describing the properties- of this interesting curve I 
shall discuss some fundamental properties of curves in general. 

* With A or A' as centre, and radius equal to OF = c, describe a circle cutting 
the y-axis at the points B, B'. Complete Fig. 22. Hence c 2 = a 2 + 6 2 . Note. For 
greater clearness in the drawing, F and F 1 have been removed a little further from 
the curve than their real position. 

F 



82 



HIGHER MATHEMATICS. 



38. 



38. A Study of Curves. 

(1) The tangent to a curve (footnote, page 77). Let OPQ be a 

curve situated, with respect to 
a pair of coordinate axes, as 
shown in Fig. 23. Let P and 
Q be two points on the curve, 
PM and QN their perpendicu- 
lars on Ox. Let PR be drawn 
parallel to MN. Join PQ and 
produce QP to cut Ox pro- 
duced at T. If Q is supposed 
to travel along the curve until 
it approaches infinitely near to 
the point P, the chord PQ be- 
FlG * 23> comes, at the limit, the tangent 

to the given curve at P. Hence the limit of the ratio RQ/PR is 

a tangent to the given curve. Or 




= Lt tan RPQ = Lt tan NTP . 



(1) 



Take any point P(x, y) on the curve POP' represented by the 
equation 

*/=/(*) . (2) 

Let the coordinates of P be increased by any arbitrary increments 
Bx and By, so that the particle occupies a new position, 

Q(x + Bx, y + By). 

OM = x ; PR = MN = to ; ON - x + Bx 
MP = y ; QR = By ; QN = QR + RN = QR + PM = y + By. 
Since the point Q also lies on the curve, 

y + By = f(x + Bx) (3) 

and EQ = By = f(x + Bx) - f(x). 

RQ _ f(x + Bx) - f(x) _ filler, y) 

' ' PR ~ &x = Bx (incr. x)' 



or di//dic = tan a .... (4) 

This is a most important result. In words, the tangent of the 
angle made by the slope of any part of the curve towards the x-axis 
is the first differential coefficient of the ordinate of the curve with 
respect to the abscissa. This rule applies to any curve. 



38. COORDINATE OR ANALYTICAL GEOMETRY. 83 

EXAMPLES. (1) Find the tangent of the angle (a) made by any point 
P(x, y) on the parabolic curve. In other words, it is required to find a 
straight line which has the same slope as the curve has which passes 
through the point P(x, y). Since 

t/ 2 = 4ax ; dyjdx - lafy = tan a. 

If the tangent of the angle were to have any particular value, this value would 
have to be substituted in place of dy/dx. For instance, let the tangent to the 
point P(x, y) make an angle of 45. Since tan 45 = unity, 

2afy = tan a = 1, .'. y = 2a. 
Substituting in the original equation y 2 = lax, we get 

x = a, 

that is to say, the required tangent passes through the extremity of the 
ordinate perpendicular on the focus. If the tangent had to be parallel to the 
.r-axis, tan being zero, dy/dx is equated to zero ; while if the tangent had to 
be perpendicular to the .r-axis, since tan 90 = oo, dyjdx oo. 

(2) Required the direction of motion at any moment of a point moving 

according to the equation, y = a cos 2*Y ^ + e V The tangent at any time t 



2ira . / t \ 
has the slope, -- -y- sin 2*-l ^ + 6 ) 



(2) Equation of the tangent line. Let TP (Fig. 24) be a tan- 
gent to the curve at the point P(x^ y^. Let OM = x v PM = y r 
Let y = mx, be the equation of the tangent line, and y l = /(#i) 
the equation of the curve. The condition that a straight line may 
pass through the point P(x v yj, is (equation (14), page 72) that 

y - 2/1 = m (% - #1) (5) 

where ra is the tangent of the angle which the line y mx makes 
with the z-axis. But we have just seen that this angle is equal 
to the first differential coefficient of the ordinate of the curve ; 
hence by substitution 

j, - y, - gl(* - *,), ... (6) 

which is the required equation of the tangent to a curve at a 
point whose coordinates are x v y r 

EXAMPLE. Required the equation of the tangent to a parabola. Since 

yi* = 40^, dy l /dx l = 2ajy r 
Substituting in (5) and rearranging terms, 

yy\ - y\ = 2 (# - -^i)- 

Substituting for y/j 2 , we get 

yy^ = 2a(x + xj 

as the equation for the tangent line of a parabola. If x 0, tan = 00, and 
the tangent is perpendicular to the x-axis and touches the ?/-axis. To get the 
point of intersection of the tangent with the x-axis put y = 0, then x = - x r 
The vertex of the parabola therefore bisects the .r-axis between the point of 
intersection of the tangent and of the ordinate of the point of tangency. 



84 



HIGHER MATHEMATICS. 



38. 



(3) Equation of the normal line. A normal line is a perpen- 
dicular to the tangent at a given 
point on the curve, drawn to the 




Let NP be normal to the curve 
(Fig. 24) at the point P(x lt yj. 
Let y = mx, be the equation to 
the normal line, y l = /(a^), the 
equation to the curve. The con- 
dition that any line may be per- 
pendicular to the tangent line TP, 
is that m' = - l/m (equation (17), page 73). From (5) 

1, 



FIG. 24. 



y - 



or 



m 
d Xl 



(7) 



(4) Equation of the subnormal. The subnormal of any curve 
is that part of the #-axis lying between the point of intersection of 
the normal and the ordinate drawn from the same point on the 
curve. 

Let MN be the subnormal of the curve shown in figure 24, then 

MN = x - x r 
The corresponding value for the length of the subnormal is, from (7), 

the normal being drawn from the point P(x lt y^. 

(5) Equation of the subtangent. The subtangent of any curve 
is that part of the #-axis lying between the points of intersection 
of the tangent and the ordinate drawn from the given point. 

Let MT (Fig. 24) be the subtangent, then 

x l - x = MT. 

Putting y = in equation (6), the corresponding value for the 
length of the subtangent is 

MT = x l - x = y^dx^dy-^. ... (9) 



(6) The length of the tangent and of the normal. The length 
of the tangent can be readily found by substituting the values 
PM and TM in the equation for the hypotenuse of a right-angled 



39. COORDINATE OR ANALYTICAL GEOMETRY. 85 

triangle TPM (Euclid, i., 47) ; and in the same way the length of 
the normal is obtained from the known values of MN and PM 
already deduced. 

EXAMPLES. (1) Find the length of the subtangent and subnormal lines 
in the parabola, y 2 = ax. Since 



the subtangent is 2x, the subnormal 2a. 

(2) Show that the subtangent of the curve pv = constant, is equal to - p. 

39. The Parabola (resumed). 

Returning now to the special curves, let P(x, y) be a point on the para- 
bolic curve (Fig. 25) referred to the coordinate axes Ox, Oy ; PT a tangent at 
the point P. Let F be the focus of the parabola y"- = 4ar. Join PP. Draw 
KP parallel to Ox. Join KT. Then KPFT is a rhombus (Euclid, i., 34), 
for it has been shown that the vertex of the parabola A bisects the subtangent 
(Example (1) above). Hence, 

TA = AM, and, by definition, OA = AF, 
.-.TO = FM, and KP = TF, 

.'. the sides KT and PF are parallel, and by definition of the parabola, 
KP = PF; .-.the two triangles KPT and PTF are equal in all respects, and 
(Euclid, i., 5) the angle KPT = angle 
TPF, that is to say, the tangent to tJie 
parabola at any given point bisects the 
angle made by the focal radius and 
the perpendicular dropped on to the 
directrix from the given point. 

In Fig. 25, the angle TPF = 
angle TPK = opposite angle RPT' 
(Euclid, i., 15). But, by construc- 
tion, the angles TPN and NPT' are 
right angles ; take away the equal 
angles TPF and RPT' and the angle 
FPNis equal to the angle NPR, that 
is to say, the normal at any point on 




FIG. 25. The Focus of the Parabola 
(after Nernst and Schonflies). 



the parabola, bisects the angle enclosed by tlie focal radius and a line draicn 
through the given point, parallel to tlie x-axis. 

This property is of great importance in physics. All light rays falling 
parallel to the principal (or x-) axis on to a parabolic mirror are reflected at 
the focus F, and conversely all light rays proceeding from the focus are re- 
flected parallel to the x-axis. Hence the employment of parabolic mirrors 
for illumination and other purposes. In some of Marconi's recent experi- 
ments on wireless telegraphy, electrical radiations were directed by means of 
parabolic reflectors. Hertz, in his classical researches on the identity of light 
and electro-magnetic waves, employed large parabolic mirrors, in the focus of 
which a " generator," or " receiver " of the electrical oscillations was placed. 
See the translation of Hertz's Electric Waves, by Jones (1893), page 172. 



HIGHER MATHEMATICS. 



40. 



SO. The Ellipse (resumed). 

The deduction of an equation for tJie tangent at any point on the ellipse is 
a simple exercise on equation (6), page 83, 



Differentiating the equation of the ellipse, x-fja 2 + y^/b 2 = 1, we obtain 

*. = _ ? a . ( 2) 

dx t a 2 y-i 
substituting this value of dy^dx-^ in (1) 

Wp. 

y-y, = -^-^ 

Multiply by y l and divide through by ft 2 , rearrange terms and combine the 
result with the equation to the ellipse. The result is the tangent to any point 
on the ellipse, 

xx-. yui 

Ir + TF^' < 3 > 

where x^ y^ are coordinates of any point on the curve and x, y the coordi- 
nates of the tangent. 

Now the tangent cuts the #-axis at a point where y = 0. Hence 

xx^ = a 2 , or, x = a 2 /x l (4) 

In Fig. 26 let PT be a tangent to the ellipse, PN the normal. From (4) 
= x + c = a z lx-, + c, FT = x - c = a 2 lx l - c, 



and 



fT 



From equations (5), page 78, 

PF l 
- 



a - cx 1 



a? + 



(6) 



From (5) and (6), therefore, 



F^T-.FT ^F^P-. 




FIG. 26. The Foci of the Ellipse (after Nernst and Schonflies). 

By Euclid, vi., A: "If, in any triangle, the segments of the base produced 
have to one another the same ratio as the remaining sides of the triangle, 
the straight line drawn from the vertex to the point of section bisects the 
external angle ". Hence in the triangle FPF^ the tangent bisects the ex- 



n. COORDINATE OR ANALYTICAL GEOMETRY. 87 

ternal angle FPR, and the normal bisects the angle FPP\. That is to say, 
the noniml tit tin if point oil the ellipse bisects ttie angle enclosed by tlie focal 
radii ; and the tangent at any point on the ellipse bisects the exterior angle 
formed by the focal radii. 

This property accounts for the fact that if F^P be a ray of light emitted 
by some source F lt the tangent at P represents the reflecting surface at that 
point, and the normal to the tangent is therefore normal to the surface of 
incidence. From a well-known optical law, " the angles of incidence and re- 
flection are equal," and since F^PN is equal to NPF when PF is the reflected 
ray, all rays emitted from one focus of the ellipse are reflected and concen- 
trated at the other focus. This well-known physical phenomena applies to 
light, heat, sound and electro-magnetic waves. 

The questions raised in 39 and 40 are treated in any textbook on 
physical or geometrical optics. 



41. The Hyperbola (resumed). 

(1) The equation of the tangent at any point P(x lt y^ on the hyperbolic 
curve, is obtained, as before, by substituting the first differential coefficient 
of the tangent to the curve in the equation 



By differentiation of the equation z 2 /a 2 - ?/ 2 /& 2 = 1, we get, 



Multiply this equation by y, divide by 6 2 , rearrange the terms and combine 
the result with (2).> We thus find that the tangent to any point on the 
hyperbola has the equation 

**i yy\ , 

-& ' -& = 1 ..... & 

At the point of intersection of the tangent with the .r-axis, y = and the 
corresponding value of x is 

= a/* 1 , ...... (4) 

the same as for the ellipse. 

The limiting position of the tangent to the point on the hyperbola at an 
infinite distance away is interesting. Such a tangent is called an asymptote. 

From (4) if x^ is infinitely great, x 0, and the tangent then passes 
through the origin. 

(2) To find the, angle which the asymptote makes with the x-axis we must 
determine the relative value of 



Multiply both sides by 6 2 /x 2 , and 



88 



HIGHER MATHEMATICS. 



If x be made infinitely great the desired ratio is 

rt y* &2 . r 

J-llX = 00 -^ 2= , . . J-J 

x 2 a 2 
Substituting this in equation (2) above we get, by writing x for x lt y for y lt 

dy a b* b 

dx = tana (say) = 5. - 2 = ^ . . . . 



(5) 



If we now construct the rectangle BSS'R' (Fig. 22, page 80) with sides 
parallel to the axis and cut off OA = OA' = a, OB = OB' - b, the diagonal 
in the first quadrant and the asymptote, having the same relation to the two 
axes, are identical. Since the x- and ?/-axes are symmetrical, it follows that 
these conditions hold for every quadrant. See page 137 for a further dis- 
cussion on the properties of asymptotes. 

42. The Rectangular or Equilateral Hyperbola. 

If we put a = b in the standard equation to the hyperbola, the 

result is a special case of the hyperbola for which 

x* - f- = a*, . . . . (6) 

and from equation (2), page 496, 

tan a = 1 = tan 45, 

that is to say, each asymptote makes an angle of 45 with the x-, 

or i/-axes. In other words, the asymptotes bisect the coordinate 

axes. This special form of the 
hyperbola is called an equilat- 
eral or rectangular hyperbola. 
It follows directly that the 
asymptotes are at right angles 
to each other. The asymptotes 
may, therefore, serve as a pair 
of rectangular coordinate axes. 
This is a very important prop- 
erty of the rectangular hyper- 
bola. 

PIG. 27.-The Rectangular Hyperbola. To fi nd the Aquation of a 

rectangular hyperbola referred 

to its asymptotes as coordinate axes. This problem is most simply 
treated as one of transformation of coordinates from one system 
(page 74) to another inclined at an angle of 45 to the old set, but 
having the same origin. 

On page 75 it was shown that if the coordinates of a point 
P(x, y) referred to one set of axes, become x and y l when referred 
to a new set, the equations of transformation are 

x x l cos a - y l sin a ; y x l sin a + y l cos a . (7) 




COORDINATE OR ANALYTICAL GEOMETRY. 89 

As shown in Fig. 27, the old axes yOx have to be rotated through 
an angle of - 45.* 

But sin ( - 45) = - 1 / ^2 ; cos ( - 45) = 1/^2 (page 497). 
Hence from (7) above 

x-xJJZ + yJJZ; y = - xj J2 + y, / ^ . (8) 
By addition and subtraction 

x - y = 2^ / ^2 ; x + y = 2^ / ^2 . . (9) 
If P(x, y) be any point on the rectangular hyperbola 

x 2 - y 2 = a 2 , or (x - y) (x + y) = a 2 . 
Substituting these values of (x - y) and (x + y) in (9), we get 



or, writing the constant term a 2 /2 = *, x for x 19 y for y lt 

xy = constant = K . . . (10) 

What is true of any point on the hyperbola is true for all points, 
that is to say, equation (10) is the equation for a rectangular 
hyperbola referred to its asymptotes as coordinate axes. 

From (10) y = K/X, and it follows that as y becomes smaller, x 
increases in magnitude. When y = 0, x = oo, that is to say, the 
,z-axis touches the hyperbola an infinite distance away. The 
same thing may be said of the ^-axis. 

43. Illustrations of Hyperbolic Curves. 

(1) The graphical representation of the gas equation, 

pv = RO, 

furnishes a rectangular hyperbola \vhen 6 is fixed or constant. 
The law as set forth in the above equation shows that the volume 
of a gas (v) varies inversely as the pressure (p) and directly as the 
temperature (0). For any assigned value of 0, we can obtain a 
series of values of p and v. For the sake of simplicity, let the 
constant R = 1. Then if 



e = 


1 


{ 


p 

v 





o-i 

10-0 


, o 

2 


5, 

o, 


1-0, 
1-0, 


5-0, 
0-2, 


10-0, . 
0-1, . 


. ; 






c 


P 


= 


0-1 


, o 


5, 


i-o, 


5-0, 


10-0, . 


* > 


e = 


5 


1 


V 


= 


5-0 


1 


o, 


0-5, 


o-i, 


0-05, . 


. . etc. 


The 


" curves 


' of 


constant 


temperature 


obtained by 


plotting 



* The trignometrical convention with regard to sign is that if a point rotates in 
the opposite direction to the hands of a watch, it is positive, if in the same direction, 
negative. 



90 



HIGHER MATHEMATICS. 



43. 



these numbers are called isothermals. Each isothermal (i.e., 
curve at constant temperature) is a rectangular hyperbola obtained 
from the equation 

pv = EB = constant, . . . (11) 
similar to (10) above. 

A series of isothermal curves, obtained by putting successively 
equal to lt 2 , 3 . . . and plotting the corresponding values of 
p and v, is shown in Fig. 28. 




FIG. 28. Isothermal _py-curves. 

We could have obtained a series of curves from the variables p 
and 6, or v and 6, according as we assume v or p to be constant. 
If v be constant, the resulting curves are called isometric lines, 
or isochores ; if p be constant the curves are isopiestic lines, or 
isobars. For van der Waals' equation, see page 398. 

(2) Exposure formula for a thermometer stem. When a ther- 
mometer stem is not exposed to the same temperature as the 
bulb, the mercury in the exposed stem is cooled, and a small 
correction must be made for the consequent contraction of the 
mercury exposed in the stem. If x denotes the difference between 
the temperature registered by the thermometer and the tempera- 
ture of the exposed stem, y the number of thermometer divisions 
exposed to the cooler atmosphere, then the correct temperature 



43. COORDINATE OR ANALYTICAL GEOMETRY. 



can be obtained by the so-called exposure formula of a thermometer, 
namely, 

= 0-00016sy, . . (12) 

which has the same form as equation (10). By assuming a series 
of suitable values for (say Ol . . . ) and plotting the result 
for pairs of values of x and y, curves are obtained for use in the 
laboratory. These curves allow the required correction to be seen 
at a glance (see Ramsay, Chemical Theory, 1893, 11). 

(3) Dissociation isotherm. Gaseous molecules under certain 
conditions dissociate into simpler parts. Nitrogen peroxide, for 
instance, dissociates into simpler molecules, thus : 



Iodine at a high temperature does the same thing, I 2 becoming 27. 
In solution a similar series of phenomena occur, KCl becoming 
K + Cl, and so on. Let x denote the number of molecules of an 




FIG. 29. Dissociation Isotherm (after Nernst and Schonflies). 

acid or salt which dissociates into two parts called ions, (1 - x) 
the number of molecules of the acid, or salt resisting dissociation, 
c the quantity of substance contained in unit volume, that is the 
concentration of the solution. Nernst has shown that at constant 
temperature 

K = c^ x .... (13) 

where K is the so-called dissociation constant whose meaning is 
obtained by putting x = 0'5. In this case K = ^c, that is to say, 



92 



HIGHER MATHEMATICS. 



43. 



K is equal to half the quantity of acid or salt in solution when 
half of the acid or salt is dissociated. 

Putting K = 1 we can obtain a series of corresponding values 
of c and x. For example, if 

x = -16, 0-25, 0-5, 0-75, 0'94 . . . ; 
then c = 32, 12, 2, 0'44, 0*07 . . . 

It thus appears that when the concentration is very great, the 
amount of dissociation is very small, and vice versa, when the 
concentration is small the amount of dissociation is very great. 
Complete dissociation can perhaps never be obtained. The graphic 
curve (Fig. 29), called the dissociation isotherm (Nernst), is 
asymptotic towards the two axes, but when drawn on <a small 
scale the curve appears to cut the ordinate axis. 

(4) The volume elasticity of a substance is defined as the ratio 

of any small increase of pres- 
sure to the diminution of 
volume per unit volume of 
substance. If the tempera- 
ture is kept constant during 
the change, we have iso- 
thermal elasticity, while 
if the change takes place 
without gain or loss of heat, 
adiabatic elasticity. If unit 
volume of gas (v) changes 
by an amount dv for an in- 
crease of pressure dp, the 
elasticity (E) is 
dp 
dv 




FIG. 30. w-curves. 



(14) 



A similar equation is obtained by differentiating Boyle's law for 
an isothermal change of state, 

pv = constant, .... (15) 

or p = - v^ (16) 

dv 

an equation identical with that deduced for the definition of volume 
elasticity. Equation (16) is that of a rectangular hyperbola re- 
ferred to its asymptotes as axes. 

Let P(p, v) (Fig. 30) be a point on the curve pv = constant. 
From the construction of figure 30, the triangles KNP and PMT 



44. COORDINATE OR ANALYTICAL GEOMETRY. 93 

are equal and similar (Euclid, i., 26). See example (2) page 85, 
and note that KN is the vertical subtangent equivalent to - p. 
KN = - AT tan a = - v tan KPN, 
dp 

--?? 

that is to say, the isothermal elasticity of a gas in any assigned 
condition, is numerically equal to the vertical subtangent of the 
curve corresponding to the substance in the given state. 

But since in the rectangular hyperbola KN = PM , the iso- 
thermal elasticity of a gas is equal to the pressure (16). The 
adiabatic elasticity of a gas may be obtained by a similar method 
to that used for equation (14). If the gas be subject to an adia- 
batic change of pressure and volume it is known that 

pvy = constant = C (say), . . . (17) 
or log 2? + ylogi 1 = log C. 

Differentiating and arranging terms 

-- 



in other words the adiabatic elasticity of a gas is y times the 

pressure. A similar construction for the adiabatic curve furnishes 

KN:MP = KP\PT 

= y:l, 

that is to say, the tangent to an adiabatic curve is divided at the 
point of contact in the ratio y : 1. 

44. Polar Coordinates. 

Instead of representing the position of a point in a plane in 
terms of its horizontal and vertical distances along two standard 
lines of reference, it is sometimes more con- 
venient to define the position of the point 
by a length and a direction. For example, 
in Fig. 31 let the point be fixed, and Ox 
a straight line through 0. Then, the position 
of any other point P will be completely de- 
fined if (1) the length OP and (2) the angle FIG. 31. Polar Co- 
OP makes with Ox, are known. These are 

called the polar coordinates of P, the first is called the radius 
vector, the latter the vectorial angle. The radius vector is 

* From other considerations, Eq is usually written A'<f- 




HIGHER MATHEMATICS. 



44. 



generally represented by the symbol r, the vectorial angle by 0, 
and P is called the point P(r, 0), is called the pole and Ox the 
initial line. As in trignometry, the vectorial angle is measured by 
supposing the angle has been swept out by a revolving line 
moving from a position coincident with Ox to OP. It is pos- 
itive if the direction of revolution is contrawise to the motion of 
the hands of a clock. 





FIG. 32. 



FIG. 33. 



To change from polar to rectangular coordinates and vice versa. 
In Fig. 32, let (r, 6) be the polar coordinates of the point P(x, y). 
Let the angle x'OP = 9. 

First, to pass from polar to Cartesian coordinates. 
HP y OM 



(1) 



In the 



y = r sin and x = rcos$, 
which determines x and y, when r and 6 are known. 

Second, to rjass from Cartesian to polar coordinates. 
same figure 

HP y 

' 



(OP) 2 = 

tan ~ * * ; r = + 



(2) 



which determines and r, when x and y are known. The sign 
of r is ambiguous, but, by taking any particular solution for 0, 
the preceding remarks will show which sign is to be taken. 
Just as in Cartesian coordinates an equation between r and 6 
may represent one or more curves. The graph may be obtained 
by assigning convenient values to 6 (say 0, 30, 45, 60, 90 . . .) 
and determining the corresponding value of r from the equation. 



$45. COORDINATK OR ANALYTICAL (iK< >M I ; 1 liY. '.:, 

EXAMPLE. Show that the polar equations of the hyperbola and ellipse 

1 eos 2 sin 2 1 cos 2 sin 2 
are respectively ^ = -^- - ^ and ^ = fl8 -55-. 

NOTE. The parabola, ellipse and hyperbola are sometimes defined as 
curves such that the ratio of the distance of any point on the curve from 
a fixed line and from a fixed point, is constant. The ratio is called the 
eccentricity, and is denoted by the letter e, the fixed point is called the 
/oc?ts, the fixed line, the directrix. 

In Fig. 33 let OAT be directrix, F the focus, AP any curve, I'K is a 
perpendicular from P on to the directrix, PM is perpendicular from P on to 
OF produced. A is vertex of curve. Then if 

FP 
e = p-fr = constant = 1, the curve is parabolic, 

FP 

e = pj = constant- <1, the curve is elliptical, 

FP 

e = pj = constant >1, the curve is hyperbolic. 

These definitions ultimately furnish equations for the hyperbola, ellipse and 
parabola similar to those adopted above. Let FP = r, OF = p, then from 
these definitions 

PK = OF + FM = p + r cos 6, 

(3) 



which is true whether curve be hyperbolic, elliptical or parabolic. 



45. Logarithmic or Equiangular Spiral. 

Equations to the spiral curves are considerably simplified by the use of 
polar coordinates. For instance, the curve for the logarithmic spiral, though 
somewhat complex in Cartesian coordinates, is represented in .polar coordi- 
nates by the simple equation 

r = a, 
where a may have any constant value. Hence 

log 7- = 01oga. 

Let Ci, G', c' . . . (Fig. 34) be a series of points on the spiral corresponding 
to the angles 3 , 6.,. . . ., then r lt r.,, . . . will represent the corresponding 
radii vectores, so that 

log r, = 6 1 log a ; log r., = 2 log ... 
Since log a is constant, say equal to fc, 



that is, the logarithm of the ratio of the distance of any two points on the 
curve from the pole is proportional to the angle between them. If r l and r., 
lie on the same straight line, then 

- 2 = 2ir = 360, 
being the symbol used in trignometry to denote 180, 

.-. log- =2k*. 



96 HIGHER MATHEMATICS. $ 45. 

Similarly, it can be shown that if r s , r 4 . . . lie on the same straight 
line, the logarithm of the ratio of r a to r s , r 4 . . . is given by 4for, 6kv. . . . 
This is true for any straight line passing through O, that is to say, the spiral 
is made up of an infinite number of turns which extend inwards and outwards 
without limit. 

If the radii vectores OC^ OD, OE . . . OC, Od . . . be taken to repre- 
sent the number of vibrations of a sounding body in a given time, the angles 
CiOD, DOE . . . may be taken as a measure of the interval between the 




FIG. 34. Logarithmic Spiral (after Donkin). 

tones produced by these vibrations. A point travelling along the curve will 
then represent a tone continuously rising in pitch, and the curve, passing 
successively through the same line produced, represents the passage of the 
tone through successive octaves. The geometrical periodicity of the curve is 
a graphical representation of the periodicity perceived by the ear when a tone 
continuously rises in pitch. 

In the above diagram the angles C^OD, DOE . . . represent the intervals 
in the diatonic scale. The intervals 

Cj to D, F to (?, A to B are major seconds, each 61 10' 22" ; 
D to E, G to A are minor seconds, each 54 43' 16" ; 

E to F, B to C are diatonic semitones, each 33 31' 11" 

(Donkin's Acoustics, page 26). 

This diagram may also be used to illustrate the Newlands-Mendeleeff law 
of octaves, by arranging the elements along the curve in the order of their 
atomic weights. 

EXAMPLES. (1) Plot Archimedes' spiral, r a6. . . . (4) 
(2) Plot the hyperbolic spiral, rO = a, . . . (5) 



4(1. COORDINATE OR ANALYTICAL GEOMETRY. 



97 




C D a. 

FIG. 35. Trilinear Coordinates. 



46. Trilinear Coordinates and Triangular Diagrams. 

Another method of representing the position of a point in a 
plane is to refer it to its perpendicular distance from the sides of a 
triangle called the triangle of reference. 
The perpendicular distances of the 
point from the sides are called tri- 
linear coordinates. In the equi- 
lateral triangle ABC (Fig. 35), let the 
perpendicular distance of the vertex A 
from the base EC be denoted by 100 
units, and let P be any point within the 
triangle whose trilinear coordinates are 
Pa, Pb, PC, then 

Pa + Pb + PC = 100. 
This property has been extensively used in the graphic repre- 
sentation of the composition of certain ternary alloys, and mixtures 
of salts. Each vertex is supposed to represent one constituent of 
the mixture. Any point within the triangle corresponds to that 
mixture whose percent- 
age composition is repre- 
sented by the trilinear 
coordinates of that point. 
Any point on a side of 
the triangle represents a 
binary mixture. Fig. 
36 shows the melting 
points of ternary mix- 
tures of isomorphous 
carbonates of barium, 

strontium and calcium. 

Such a diagram is some- BC0 3 SrC 3 

times called a surface of FlG - 36. Surface of Fusibility. 

fusibility. A mixture melting at 670 may have the composition 
represented by any point on the isothermal curve marked 670, 
and so on for the other isothermal curves. 

In a similar way the composition of quaternary mixtures has 
been graphically represented by the perpendicular distance of a 
point from the four sides of a square. 

Roozeboom, Bancroft and others have used triangular diagrams 

G 




98 HIGHER MATHEMATICS. 46. 

with lines ruled parallel to each side as shown in Fig. 37. Suppose 
we have a mixture of three salts, A,B, C, such that the three vertices 
of the triangle ABC represent phases * containing 100 / of each 
component. The composition of any binary mixture is given by a 
point on the boundary lines of the triangle, while the composition of 
any ternary mixture is represented by some point inside the triangle. 
The position of any point inside the triangle is read directly from 
the coordinates parallel to the sides of the triangle. For instance, 
the composition of a mixture represented by the point is given 




p o 

FIG. 37. Concentration-Temperature diagram (after Bancroft). 

by drawing lines from 0, parallel to the three sides of the triangle 
OP, OR, OQ. Then start from one corner as origin and measure 
along the two sides, AP fixes the amount of C, AQ the amount of 
B, and, by difference, GE determines the amount of A. For the 
point chosen, therefore A = 40, B 40, C = 20. 

(1) Suppose the substance A melted at 320, B at 300, and C 
at 305, and that the point D represents an eutectic alloy j- melting 

* A phase is a mass of uniform concentration. The number of phases in a system 
is the number of masses of different concentration present. For example, at the tem- 
perature of melting ice three phases may be present in the /f 2 0-system, viz., solid ice, 
liquid water and steam ; if a salt is dissolved in water there is a solution and a vapour 
phase, if solid salt separates out, another phase appears in the system. 

f An eutectic alloy is a mixture of two substances in such proportions that the 
alloy melts at a lower temperature than a mixture of the same two substances in any 
other proportions. The numbers chosen are based on Guthrie's work (Philosophical 
Magazine [5], 17, 462, 1884) on the nitrates of potassium (.4), lead (B), sodium (C). 



47. COORDINATE OR ANALYTICAL <;K< >.M K TRY. 99 

at 215 ; at E, A and B form an eutectic alloy melting at 207 ; at 
F, B and C form an eutectic alloy melting at 268. 

(2) Along the line DO, the system A and C has a solid phase ; 
along EO, A and B have a solid phase ; and along FO, B and C 
have a solid phase. 

(3) At the triple point 0, the system A, B and C exists in the 
three solid, solution and vapour phases at a temperature of 186 (say). 

(4) Any point in the area ADOE represents a system com- 
prising solid, solution and vapour of A, in the solution, the two 
components B and C are dissolved in A. Any point in the area 
CD OF represents a system comprising solid, solution and vapour 
of C, in the solution, A and B are dissolved in C. Any point in 
the area BE OF represents a system comprising solid solution and 
vapour of B, in the solution, A and C are dissolved in B. 

Each apex of the triangle not only represents 100 / of a 
substance, but also the temperature at which the respective 
substances A, B, or C melt ; D, E, F also represent temperatures 
at which the respective eutectic alloys melt. It follows, therefore, 
that the temperature at D is lower than at either A or C. Simi- 
larly the temperature at E is lower than at A or B, and at F 
lower than at either B or C. The temperature, therefore, rises 
as we pass from one of the points D, E, F to an apex on either side. 

For full details the reader is referred to Bancroft's The Phase 
Rule, 1897, p. 147. 

47. Orders of Curves. 

The order of a curve corresponds with the degree of its equa- 
tion. The degree of any term may be regarded as the sum of the 
exponents of the variables it contains ; the degree of an equation 
is that of the highest term in it. For example, the equation 
xy + x + b z y = 0, is of the second degree if b is constant ; the 
equation ar { + xy = 0, is of the third degree ; x z yz z + ax = 0, is of 
the sixth degree, and so on. 

1st. A line of the first order is represented by the general 
equation of the first degree 

ax + by + c = . . . . (1) 
This equation is that of a straight line only. 

2nd. A line of the second order is represented by the general 
equation of the second degree between two variables, namely, 

ax 2 + %hxy + by* + %gx + %fy + c = . . (2) 



100 HIGHER MATHEMATICS. 47. 

This equation includes, as particular cases, every possible form of 
equation in which no term contains x and y as factors more than 
twice. The term 2hxy can be made to disappear by changing the 
direction of the rectangular axes, and the terms containing 2gx 
and 2/2/ can be made to disappear by changing the origin of the 
coordinate axes. Every equation of the second degree can be 
made to assume one of the forms 

ax* + by 2 = q, . (3) 

or y 2 = px . . . . . (4) 

The first can be made to represent a circle,* ellipse, or hyperbola ; 
the second a parabola. Hence every equation of the second degree 
between two variables includes four species of curves circle, ellipse, 
parabola and hyperbola. 

It must be here pointed out that if two equations of the first 
degree with all their terms collected on one side be multiplied 
together we obtain an equation of the second degree which is 
satisfied by a quantity which satisfies either of the two original 
equations. An equation of the second degree may thus represent 
two straight lines, as well as one of the above species of curves. 
The same thing applies for equations of the third and higher 
degrees. 

The condition that the general equation of the second degree may represent 
two straight lines. Rearranging the terms of equation (2) in descending 
powers of y, we obtain 

by* + 2(fa; + f)y + ax 2 + 2gx + c = 0. 
This may be solved like a quadratic in y with the result that 



y= N/{(ft a - ab)x* + 2(hf - bg)x + (f* - 6c)}/6 - (hx + f)[b (5) 
an expression analogous to the tangent form of the equation to a straight line, 

y = mx + b. 

The two solutions in equation (5) can only represent straight lines if the 
quantity under the root sign can be extracted as a simple rational expression 
in x, that is to say, if 

(h* - ab)x z + 2(fe/ - bg)x + (f* - be) 
is a perfect square (see page 388). This condition is satisfied when 

(hf - bg)* = (h* - ab) (/ - be). 
Multiplying out and dividing by b (when b is not zero) 

abc + 2fgh - af* - bg* - cW = 0. . . . (6) 
If b = and a is not zero, we obtain the same result by solving for x. 
If a = b = 0, resolve the original expression into factors and 

(x + flh) (y + gjh) = 0, 
provided c/2fe = fg/W, or 2fgh - cW - 0. This agrees with equation (6). Under 

* The circle may be regarded as an ellipse with major and minor axes equal. 



48. COORDINATE OR ANALYTICAL GEOMETIIY. 101 

these circumstances equation (2) represents two straight lines respectively 
parallel to the two axes. 

The general equation of the second degree may represent a parabola, ellipse, 
or Jujperbola, according as h? - ab, is zero, negative, or positive. 

EXAMPLE. Show that the equation 

S.r a - 3xy - 3y* + 5f + 4y + 4 = 0, 
represents two straight lines as well as an ellipse. 

3rd. A line of the third order is represented by the general 
equation of the third degree between two variables 

ay* + by 2 x + cyx 2 + fx* + . . . + n = . (7) 
Sir Isaac Newton has shown that some eighty species of lines are 
included in this equation ; these may be reduced to one of the 
following four classes : 

a# 3 + bx 2 + ex + / = xy 2 + gy . . (8) 

ax* + bx 2 + ex + / = xy . . . . (9) 
ax 3 + bx 2 + ex + f = y 2 . . . . (10) 
ax 3 + bx 2 + ex + f = y . . . . (11) 

The last (equation 11) includes the cubical parabola i/ 3 = ax. 

EXAMPLES. The student will gain more information by plotting all these 
curves on squared paper, than by reading pages of descriptive matter. Use 
table of cubes, page 517. 

4th. A line of the fourth order is represented by the general 
equation of the fourth degree between two variables, viz., 

ay 4 + by*x + cy 2 x 2 + fyx 3 + . . . + t = . (12) 
Euler divided these into some 200 species which reduce to 146 
classes. At the present time the number of species is said to 
exceed 5,000. 

A family of curves is an assemblage of curves defined by one equation of 
an indeterminate degree. For example, the number of parabolas whose abscis- 
sae are proportional to any power of the ordinates is infinite. Their equation is 

y n = ax. 

For the common or quadratic parabola n 2, for the cubic parabola n = 3, 
and for the biquadratic parabola n = 4. 

The study of curves of higher orders than the third is perhaps 
more interesting than useful, at least so far as practical work is 
concerned. 

48. Coordinate Geometry in Three Dimensions. Geometry 

in Space. 

(1) The graphic representation of functions of three variables. 
Methods have been described for representing changes in the state 



102 HIGHER MATHEMATICS. 48. 

of a system involving two variable magnitudes, by the locus of a 
point moving in a plane according to a fixed law defined by the 
equation of the curve. Such was the _pv-diagram described on 
page 90. There, a series of isothermal curves were obtained, 
when was made constant during a set of corresponding changes 
of p and v in the equation 

pv = EO, 
where H is constant. 

When any three magnitudes, x, y, z, are made to vary together, 
we can, by assigning arbitrary values to two of the variables, find 
corresponding values for the third, and refer the results so ob- 
tained to three fixed and intersecting planes called the coordinate 
planes. The lines of inter- section of these planes are the 

coordinate axes. Of the re- 
sulting eight quadrants, four 
of which are shown in Fig. 38, 
only the first is utilised to any 
great extent in mathematical 
physics. This mode of graphic 
representation is called geome- 
try in space, or geometry in 
three dimensions. 

If we get a series of sets of 
corresponding values of x, y, z 
in the equation 

FIG. 38. Cartesian Coordinates in x + y = z, 

Three Dimensions. -, , ,-, .-,. 

and refer them to coordinate 

axes in three dimensions, as described below, the result is a plane 
or surface. If one of the variables remains constant, the result- 
ing figure is a line. A surface may, therefore, be considered to be 
the locus of a line moving in space. 

The position of the point P with reference to the three co- 
ordinate planes xOy, xOz, yOz (Fig. 38) is obtained by dropping 
perpendiculars PL, PM, PN from the given point on to the three 
planes. Complete the parallelepiped, as shown in Fig. 38. Let 
OP be a diagonal. Then LP = OA, NP = OB, HP = OC. 

To find the point luhose coordinates OA, OB, OC are given. 
Draw three planes through A, B, C parallel respectively to the 
coordinate planes ; the point of intersection of the three planes, 
namely P, will be the required point. 




4S. 



COORDINATE OR ANALYTICAL GEOMETRY. 




If the coordinates of P, parallel to Ox, Oy, Oz, are respectively 
x, y and z, then P is said to be the point x, y, z. A similar 
convention with regard to the 
sign is used as in analytical geo- 
metry of two dimensions, with 
the additional convention that y 
is positive when in front of ; 
negative, when behind. It is 
necessary that the reader shall 
have a clear idea of spatial geo- 
metry in working many physical 
problems. 

(2) To find the distance of a ^/ M 
point from the origin in terms of 

the rectangular coordinates of that point. In Fig. 39, let Ox, Oy, Oz 
be three rectangular axes, P(x, y, z) the given point such that 
PM = z, MA = y, OA = x. It is required to find the distance 
OP = r, say. From the construction (rectangular coordinates) 

OP 2 = OM 2 + PM 2 , or r 2 = OM 2 + z 2 , 
but OM' 2 = MA' 2 + OA 2 = x 2 + y 2 . 

r 2 = x 2 + f' + z' 2 (1) 

Let the angle POx = a ; POy = ft ; POz = y, then 

x = r cos a ; y = r cos ft ; z = r cos y . (2) 

These equations are true wherever the point P may lie, and 
therefore the signs of x, y, z are always the same as those of cos a, 
cos ft, cos y respectively. Substituting these values in (1), and 
dividing through by r 2 , we get the 
following relation between the three 
angles : 

COS 2 a + COS' 2 ft + COS' 2 y =1. (3) 

These cosines are called the direc- 
tion cosines, and are usually sym- 
bolised by the letters /, m, n. Thus 
(3) becomes 

(3) To find the distance between 

two points in terms of their red- FlQ - 40 - 

angular coordinates. Let P^x^ y lt zj, P 2 (# 2 , y. 2 , z 2 ) be the given 
points, it is required to find the distance PjP 2 in terms of, the 
coordinates of the points P l and P . Draw planes through P l 




104 HIGHER MATHEMATICS. 48. 

and P 2 parallel to the coordinate planes so as to form the parallele- 
piped ABODE. By the construction (Fig. 40), the angle P 1 EP 2 
is a right angle. Hence 

PjP 2 2 = Pj# 2 + P 2 E* = P^ 2 + DE* + P 2 D*. 
But P^ is evidently the difference of the distance of the foot of 
the perpendiculars from P l and P 2 on the a?-axis, or P l E = x 2 -x l . 
Similarly DE = y 2 - y lt P 2 D = z 2 - z r Hence 

^ r 2 = (x 2 - x,Y + (y, - y,Y + (** - *i) 2 . (5) 

(4) To find the angle between two straight lines whose direction 
cosines are given. In the preceding diagram (Fig. 40) join OP 1 
and OP 2 . Let \f/ be the angle between these two lines. In the 
triangle P 2 OP l (formula 47, page 500) if OP l = r l , OP 2 = r 2 , P X P 2 = r, 

r 2 = r x 2 + r 2 2 - 2^ cos if/. 
Eearranging terms and substituting 

r x 2 = x* + y, 2 + V ; r 2 2 = x* + y* + 2 2 , 
we get 

cos ^ = (x^ + yM 2 + ^^ 2 )/ r i r 2- 
Substituting, as in (2), 

x l = r x cos aj ; x 2 = r 2 cos a 2 ; y 2 = r 2 cos (3 2 . . . 

COS l// = COS ttj . COS a 2 + COS (3 l . COS y8 2 + COS y x . COS y 2 (6) 

or, from (4), 

COS \jr = IJ 2 + WjWg + ?^W 2 , ... (7) 

which represents the angle between two straight lines whose 
direction cosines are known. 

If the lines are perpendicular, cos \f/ cos 90 = 0. Hence 

COS ttj . COS a 2 + COS /^ . COS /3 2 + COS y 1 . COS y 2 = (8) 

IJ 2 + m^m^ + n-^n.2 = . . (9) 

If the vectors r lt r 2 (page 93) are known, multiply (6) by r^, 
and, remembering that 

r x cos a x = x 1 ', r 2 cos a 2 = x 2 ; r 2 cos y 2 = z 2 , etc., 
we may write a preceding result : 

* rfa cos ^ = x^ 2 + y$ 2 + z^ t . (10) 

and when the lines are perpendicular, 



(5) Projection. If a perpendicular be dropped from a given 
point upon a given plane the point where the perpendicular touches 
the plane is the projection of the point P upon that plane. For 
instance, in Fig. 38, the projection of the point P on the plane 
xOy is M, on the plane xOz is N, and on the plane yOz is L. 
Similarly, the projection of the point P upon the lines Ox, Oy, Oz 
is at A , B and C respectively. 



$ 48. COORDINATE OR ANALYTICAL GEOMETRY. 105 



In the same way the projection of any curve on a given plane 
is obtained by projecting every point in the curve on to the plane. 
The plane, which contains all the perpendiculars drawn from the 





FIG. 41. Projecting Plane. 



FIG. 42. 




FIG. 43. 



different points of the given curve, is called the projecting plane. 
In Fig. 41, CD is the projection of AB on the plane EFG ; ABCD 
is the projecting plane. 

EXAMPLES. (1) The pro- 
jection of any given line on 
an intersecting line is equal 
to the product of the length of 
the given line into the cosine 
of the angle of intersection. 
In Fig. 42, the projection of 
AB on CD is AE, but AE = 
AB cos 6. 

(2) In Fig. 43, show that 
the projection of OP on OQ 
is the algebraic sum of the 
projections of OA, AM, MP, 
taken in this order, on OQ. Hence, if OA = x, OB = AM = y, OC = PM = z 
and OP = r, from (6) 

r cos ^ = x cos a + y cos + z cos 7 . . . (12) 

(6) To find the equa- 
tion of a plane surface in 
rectangular coordinates. 
Let ABC (Fig. 44) be the 
given plane whose equa- 
tion is to be determined. 
Let the given plane cut 
the coordinate axes at 
points A, B, C, such that 
OA = a, OB = 6, OC = c. 
From any point P(x, y, z) 



drop the perpendicular 




FIG. 44. 



106 HIGHER MATHEMATICS. 48. 

PM on to the yOx plane. Then OA' = x, MA' = y and PM = z. 
It is required to find an equation connecting the coordinates with 
the intercepts a, b, c. From the similar triangles AOB, A A'B', 
OA'.OB = A' A : A'B' ; or a : b = a - x : A'B', 

.'. A'B' = b - ; similarly, MB' = b - y -- . 
Again, from the similar triangles COB, C'A'B', PMB', 

7)7- 

CO : OB = PM : MB' ; or c : b = z : b - y - , 

bcx 
.-, bz = cb - cy - -. 

a 

Divide through by be and rearrange for the required : 



an equation very similar in form to that developed on page 69. 

We may write this equation in its most general form, 

Ax + By + Cz + D = 0, . . . 
which is the most general equation of the first degree between 
three variables. Equation (14) is the general equation of a 
plane surface. It is easily converted into (13) by substituting 
Aa + D = 0, Bb + D = 0, Cc + D = 0. 

If OQ = r (Fig. 44) be a perpendicular on the plane ABC, the 
projection of OP on OQ is equal to the sum of the projections of 
OM, PM, MA on OQ (see example (2), page 105). Hence 

x cos a + y cos y8 + z cos y = r . . (15) 
from (14) 

cos 2 a : cos 2 : cos 2 7 =A 2 : B 2 : C' 2 
componendo,* 

(C08 2 a + COS 2 + COS 2 y) I COS 2 a - A 2 + B' 2 + C 2 : A 2 . 

But by (3), the term in brackets is unity, 



' cosa = ffl + o ; cos/3 = 

ri 



cos = 



* If a, &, c and d are proportionals, 
a : b = c : d 

b : a = d : c (invertendo) 

a : c b : d . . . . . . (alternando) 

a + b:b = c + d:d. . . , . (componeiulo] 

a b : b c - d : d (dimdendo) 

a : a b = c : c d . . . . . (convertendo) 

a + b:a + b = c + d: c + d . . . . (componendo et dimdendo). 

(See any elementary text-book on algebra.), 



48. 



COORDINATE OK ANALYTICAL <JK< >.M KTIIY. 



107 



Dividing equation (14) through with 
from (16), 



+ ff 2 + C 2 , we get, 
D 



X COS a + y COS ft + Z COS y = - -j======^ (17) 

where - D / J(A 2 + B 1 + C' 2 ) represents the distance of the plane 
from the origin. 

If ABC (Fig. 44) represents the face, or plane of a crystal, the intercepts 
a, b, c on the #-, y- and z-axes are called the parameters of that plane. The 
parameters in crystallography are usually expressed in terms of certain axial 
lengths assumed unity. If OA = a, OB = b, OC = c, any other plane, whose 
intercepts on the j>, y- and 2-axes are respectively p, q and r, is defined by 
the ratios 

a b c 
p ' q ' r' 

These quotients are called the parameters of the new plane. The reciprocals 
of the parameters are the indices of a crystal face. The several systems of 
crystallographic notation which determine the position of the faces of a 
crystal with reference to the axes of the crystal are based on the use of 
parameters and indices. 

(7) To find the equation of a straight line in rectangular co- 
ordinates. A line in space is represented in mathematics by 
two equations. If we consider a 
straight line in space to be formed 
by the intersection of two projecting 
planes, formed, in turn, by the pro- 
jection of the given line on two 
coordinate planes, the equation to 
the straight line evidently consists of 
two parts. Let ab, a'b' be the projec- 
tion of the given line AB on the xOz 
and the yOz planes, then (Fig. 45) 

x = mz + c ; y = m'z + c'. (18) 
Here m represents the tangent of 
the angle which the projection of 
the given line on the xOz plane makes with the #-axis ; m the tan- 
gent of the angle made by the line projected on the yOz plane 
with the ?/-axis ; c is the distance intercepted by the projection of 
the given line on the #-axis ; c', a similar ihtersection on the y-axis. 

If we eliminate z from equation (18), 

y-c' = ^(x-c) . . . (19) 
represents the projection of the given line on the xOy plane. 




\ 



B 



FIG. 45. 



108 



HIGHER MATHEMATICS. 



48. 



(8) Surfaces of revolution. A surface is assumed to have been 
generated by the motion of a line in space. If the line rotates 
round a fixed axis, the rotating surface is called a surface of revo- 
lution. Thus, a sphere may be formed by the rotation of a circle 
about a diameter ; a cylinder may be formed by the rotation of a 
rectangle about one of its sides as axis ; a cone may be generated 

by the revolution of a triangle about its 
axis ; an ellipsoid of revolution, by the 
rotation of an ellipse about its major or 
minor axes ; a paraboloid, by the rotation 
of a parabola about its axis. If an hyper- 
bola rotates about its transverse axis, 
two hyperboloids will be formed by the 
revolution of both branches of the hyper- 
bola. On the other hand, only one 
hyperboloid is formed by rotating the 
hyperbolas about their conjugate axes. 
In the former case, the hyperboloid is 
said to be of two sheets, in the latter, of 
one sheet. 

(9) To find the equation of the surface 
of a right cylinder. Let one side of a 
rectangle rotate about Oz as axis. Any point on the outer edge 
will describe the circumference of a circle. If P(x, y, z] (Fig. 46) 
be any point on the surface, r the radius of the cylinder, then the 
required equation is 

r 2 = x* + y 2 . . . . (20) 

The equation to a right cylinder is thus independent of z. This 
means that z may have any value whatever assigned to it. 

EXAMPLES. (1) Show that the equation of a right coneis x 2 + y 2 - 2 2 tan 2 <J> = 0, 
where <> represents half the angle at the apex of the cone. (Origin of axes is 
at apex of cone.) 

(2) The equation of a sphere is x 2 + y 2 + z 2 = r 2 . Prove this. 

(10) To find the equation of the tangent plane * at any point of 
a surface. Let P(x lt y^z^) be any point on the curved surface, 

z=f(x,y). . (21) 

The tangent plane through the point P(x lt y lt zj will be determined 




FIG. 46. 



* The tangent plane of a surface is a plane which touches the surface at a point, or 
a line. 



COORDINATE OR ANALYTICAL GEOMETRY. 



when two linear tangents through this point have been determined. 
For the sake of simplicity, consider the linear tangents parallel to 
the xz- and the zy-pl&nes. As an exercise after (6), page 83, 
the reader will be able to show that these two tangent lines have 
equations, 

ft. 9 

(22) 



z , -^te- 
1 dx^ 



z - z l = ~(y - 2/0 ; x = x v (23) 

where the partial derivatives, dzjdx^ dzjdy^ obviously represent 
the trignometrical tangents of the lines of intersection of the tan- 
gent plane with the coordinate planes xz and zy respectively. 
Hence, the equation to this plane is, 

dz, , dz, , 



EXAMPLE. Prove that the tangent plane to the surface 

u = /(z, y, z) = 0, 
du du du 



(25> 



(11) Polar coordinates. Instead of referring the point to its- 
Cartesian coordinates in three dimen- 
sions, we may use polar coordinates. 
In Fig. 47, let P be the given point 
whose rectangular coordinates are x, y, 
z ; and whose polar coordinates are r, 
0, <, as shown in the figure. 

(i) To pass from rectangular to 
polar coordinates. 
x=OA = OMcos <f> = r sin . cos <) 

(26) 




,.. x m / i FIG. 47. Polar Coordinates in 

(n) To pass from polar to rectangu- Three Dimensions . 



lar coordinates. 



r = 




(27) 



110 



HIGHER MATHEMATICS. 
49. Orders of Surfaces. 



49. 



Just as an equation of the first degree between two variables 
represents a straight line of the first order, so does an equation of 
the first degree between three variables represent a surface of the 
first order. Such an equation in its most general form is 

Ax + By + Cz + D = 0, 
the equation to a plane. 

An equation of the second degree between three variables re- 
presents a surface of the second order. The most general 
equation of the second degree between three variables is 

Ax* + By 2 + Cz' 2 + Dxy + Eyz + Fzx + . . . + N = 0. 
All plane sections of surfaces of the second order are either circular, 
parabolic, hyperbolic, or elliptical, and comprised under the generic 
word conicoids, of which spheroids, paraboloids, hyperboloids and 
ellipsoids are special cases. 

A surface of the second degree may be formed by plotting from 
the gas equation 

f(p, v,6) = 0; or pv = BO, 

by causing p, v and 6 to vary simultaneously. The surface pabv 
(Fig. 48) was developed in this way. 





FIG. 49. 



FIG. 48. jpv0-surface. 



Since any section cut perpendicular to the 0-axis is a rect- 
angular hyperbola, the surface is an hyperboloid. The isothermals 
6, 2 , ^ 3 , ... (Fig. 28, page 90) may be looked upon as plane 
sections cut perpendicular to the #-axis at points corresponding to 
O lt 2 > > an ^ tnen projected upon the pv-pl&ne. In Fig. 49, 
the curves corresponding to pv and ab have been so projected. 






COORDINATE OR ANALYTICAL GEOMETRY. 



Ill 



If a sufficient number of such projections were available, the 
characteristic equation, f(p, v, 6) = 0, would be solved completely. 

As a general rule, the surface generated by three variables is 
not so simple as the one represented by a gas obeying the simple 
laws of Boyle and Gay Lussac. 

van der Waals "<|/" surfaces are developed by using the 
variables ty, x, v, where ^ denotes the thermodynamic potential at 
I'oM-uaiit volume (U - 0<f>) ; x the composition of the substance ; v 
the volume of the system under investigation. The "i^" surface 
is analogous to, but not identical with, pabv in the above figure. 
Full particulars are given in van der Waals' classic, Die Continuitdt 
des (jasfrrmigen und fliissiyen Zustandes, Theil II. 

The so-called thermodynamic surfaces of Gibbs are obtained 
in the same way from the variables v, U, </> (or volume, internal 
energy and entropy) of a given system. They are described with 
some detail in Preston's Theory of Heat, page 685, or better still, 
Le Chatelier's Equilibre des systemes chimiques par J. Willard 
Gibbs, p. 98 (see also page 343). * 



50. Periodic or Harmonic Motion. 

Let P (Fig. 50) be a point which starts to move from a position 
of rest with a uniform velocity on the perimeter of a circle. Let 
xOx, yOy' be coordinate 
axes about the centre O. 
Let P lf P 2 . . . be positions 
occupied by the point after 
the elapse of intervals of 
time Jj, ^ 2 ... From P l 
drop the perpendicular 
M l P l on to the #-axis. 
Remembering that if the 
direction of M^, M 2 P. 2 
... be positive, that of 
3f 3 P 3 , M 4 P 4 is negative, and 
the motion of OP as P 
revolves about the centre 
in the opposite direction 




FIG. 50. Harmonic or Periodic Motion. 



to the 
then 



hands of a clock is conventionally reckoned positive, 



112 HIGHER MATHEMATICS. 50. 

~M~P S ~M 4 P 4 

= ^oir-; sma 4 = ; FB ^ 

Or, if the circle have unit radius r = 1, 

sinaj = + M-f-i', sina 2 = + M 2 P 2 ; sina 3 = - M 3 P S ; sina 4 = - M 4 P 4 . 
If the point continues in motion after the first revolution, this series 
of changes is repeated again and again. 

During the first revolution, if we put TT = 180, and let O lt 2 ,. 
. . . represent certain angles described in the respective quadrants, 

6 l = aj ; $2 = TT a 2 ; $3 = TT + a g ; 4 = 2ir a 4 . 
During the second revolution, 

1 = 2?r + ai ; 2 = 2?r + (TT - a. 2 ) ; O z = 2?r + (TT + a 3 ), etc. 
We may now plot the curve 

y = sin a . . (1) 

by giving a series of values 0, JTT, |TT . . . to a and finding the 
corresponding values of y. Thus if 

X = a = 0, ^TT, TT, |TT, 2?r, f TT, . . . ; 

y = sin 0, sin JTT, sin TT, sin |TT, sin 2?r, sin |TT, . . . ; 
= sin 0, sin 90, sin 180, sin 270, sin 360, sin 90, . . . ; 
= 1, 0, - 1, 0, 1, 0, ... 

Intermediate values are sin JTT = sin 45 = '707, sin |TT = - -707 . . . 
The curve so obtained has the wavy or undulatory appearance 

shown in Fig. 51. It 

y is called the curve of 

sines or the har- 
monic curve. 
_ A function whose 
value* recurs at fixed 
intervals when the 
variable uniformly in- 

creases in magnitude 
FIG. 51.-Curve of Sines, or Harmonic Curve. ig ^ to be ft 




odic function. Its mathematical expression is 

f(t)=f(t + nt)^ (2) 

where n may be any positive or negative integer. In the present case 
n = 2?r. The motion of the point P is said to be a simple harmonic 
motion. Equation (1) thus represents a simple harmonic motion. 

If we are given a particular value of a periodic function of, 
say, t, we can find an unlimited number of different values of t 
which satisfy the original function. Thus 2$, 3, It, . . ., all 
satisfy equation (2). 



"><>. 



COOliniNATK OK ANALYTICAL < JKo.M KTIiY. 



EXAMPLES. (1) Show tliat the graph of y = cos a has the same form as 
the sine curve and would he identical with it if the //-axis of the sine curve 
were shifted a distance of $TT to the right. [Proof : sin ($ir + x) cos x, etc.] 
The physical meaning of this is that a point moving round the perimeter of 
the circle according to the equation y = cos a is just $*, or 90 in advance of 
one moving according to y = sin a. 

(2) Illustrate graphically the periodicity of the function y = tan a. (Note 
the passage through + oo.) 

Instead of taking a circle of unit radius, let r denote the mag- 
nitude of the radius, then 

y = r sin a. 

Since sin a can never exceed the limits + 1, the greatest and least 
values y can assume are - r and + r ; r is called the amplitiule of 
the curve. The velocity of the motion of P determines the rate at 
which the angle a is described by OP (called the aiujular velocity). 
Let t denote the time, o> the angular velocity, 
da 



Tt = "> r 



wt, 



and the time required for a complete revolution is 

t = 27T/W, 

which is called the periodic value of a, or period of oscillation, or 
periodic time ; 2?r is the wave, length. If E (Fig. 50) denotes some 
arbitrary fixed point such that the periodic time is counted from 
the instant P passes through E, the angle xOE = e, is called the 
epoch, and the angle described 
by OP in the time t = ut + e 
= a, or 

y = r sin (wt 4 e) . (3) 
Electrical engineers call the 
lead or, if negative, the lag of 
the electric current. 

EXAMPLE. Show that the 
graph of equation (3) may be re- 
presented by a curve of the form 
shown in Fig. 52. FIG. 52. 

The motion of M (Fig. 50), that is to say, the projection of the 
moving point on the diameter of the circle xOx' is a good illustra- 
tion of periodic motion, already discussed, page 48. The motion 
of an oscillating pendulum, of a galvanometer needle, of a tuning 
fork, the up and down motion of a water wave, the alternating 
electric current, sound, light, and electromagnetic waves are all 

H 





114 HIGHER MATHEMATICS. 50. 

periodic motions. Many of the properties of the chemical ele- 
ments are periodic functions of their atomic weights (Newlands- 
Mendeleef law). Some interesting phenomena have recently come 
to light which indicate that chemical action may assume a periodic 
character.* The evolution of hydrogen gas, when hydrochloric acid 

acts on one of the allo- 
tropic forms of chromi- 
um, has recently been 
studied by W. Ost- 
wald.l He found that 
if the volume of gas 
evolved during the 

action be plotted as 

FIG. 53.-Ostwald's Curve of Chemical Action. or di na te against the 

time as abscissa, a curve is obtained which shows regularly alter- 
nating periods of slow and rapid evolution of hydrogen. The 
particular form of these " waves " varies with the conditions of the 
experiment. One of Ostwald's curves is shown in Fig. 53. 

Composition of harmonic motions. It is important to remember 
that two or more simple harmonic motions may be compounded 
into one. Thus it can be shown that 

a sin (qt + e) -f b cos (qt + e) = A sin (qt + ej . (4) 
where q has the same meaning as w above. Expand the left-hand 
side of (4) according to formulae (21) and (22), page 499 ; re- 
arrange terms to obtain 

= sin qt(a cos e - b sin e) + cos qt(b cos e + a sin e) ; 
= A sin (qt + e^, 
provided 

A cos j = a cos e - b sin e ; A sin l = b cos e + sin e. (5) 
Square equations (5) and add 

A* = a 2 + b' 2 (6) 

* Abney has noticed that if a photographic film be "exposed" for a much 
longer period than is required it will after a certain interval return to a sensitive 
condition. Troost and Hautefeuille state that silicon hexachloride (Stt 2 C7 6 ) is stable 
above 1,0.00 and below 350; hydrazine hydrate (N Z H 4 . H 2 O) ; ozone (O s ), hydrogen 
selenide (H%Se); cyanogen (C 2 N 2 ) ; acetylene (C 2 H 2 ) 5 and nitrogen peroxide (A^O.,) 
are said to exhibit similar phenomena ; the action of chlorine on platinum, of oxygen 
on copper and on phosphorus is also said to be similar. Many of these statements, 
no doubt, arise from a faulty interpretation of experimental work. But the subject 
certainly merits a closer investigation. 

f W. Ostwald, Zeitschrift fur physlkalische Chemie, 35, 33, 204, 1900 ; Brauner, 
ih., 38,441, 1901. 



g 51 COORDIN v i I. OB AN \\.\ I K'\i. QEOMETBY, n:, 

Divide equations (5), rearrange terms and show that 

tan(c - l ) = -b/a, .... (7) 

from formulae (21) and (22), page 499. When c = 0, 

tan l = b'a. . . . . . (8) 

liquations (6) and (7) are the necessary conditions that (4) 
may hold good. Give a geometrical inter- 
pretation to (4), (6) and (8), by means of 
figure 54. N 




EXAMPLES. (1) Draw the graphs of the two 
curves, 

y = a sin (qt + e) and y l = a sin (qt + * j). 
Compare the result with the graph of 

y z = a sin (qt + ) + % sin (qt + ej. 
(2) Draw the graphs of 

y l = sin x, y. 2 = \ sin x, y, = i sin 5.r, y = sin x + } sin 3x + i sin 5x 
(see page 363 for this and other examples). 



51. Generalised Forces and Coordinates. 

When a mass of any substance is subject to some physical 
change, certain properties (mass, chemical composition) remain 
fixed and invariable, while other properties (temperature, pressure, 
volume) vary. When the value these variables assume in any 
given condition of the substance is known, we are said to have a 
complete knowledge of the state of the system. These variable 
properties are not necessarily independent of one another. We 
have just seen, for instance, that if two of the three variables 
defining the state of a perfect gas are known, the third variable 
can be determined from the equation 

pv = BO, 

where It is a constant. In such a case as this, the third 
variable is said to be a dependent variable, the other two, in- 
>/i']>,')ident variables. When the state of any material system 
c<tn be defined in terms of n independent variables, the system 
iid to possess n degrees of freedom, and the n independent 
rariables are called generalised coordinates. For the system 
just considered n = 2, and the system possesses two degrees of 
freedom. 

Again, in order that we may possess a knowledge of some 



116 HIGHER MATHEMATICS. 51. 

systems, say gaseous nitrogen peroxide, not only must the vari- 
ables given by the gas equation 

<f>(p, v, 0) = 

be known, but also the mass of the N 2 4 and of the N0 2 present. 
If these masses be respectively ra 1 and m. 2 , there are five variables 
to be considered, namely, 

<k(p, v, 0, m lt m. 2 ) = 0, 

but these are not all independent. The pressure, for instance, may 
be fixed by assigning values to v, 0, m lt m 2 ; p is thus a dependent 
variable, v, 0, m l} m 2 are independent variables. Thus 

p = f(v, 6, m lt m z ). 

We know that the dissociation of N. 2 into %N0. 2 depends on the 
volume, temperature and amount of N0 2 present in the system 
under consideration. At ordinary temperatures 

m i = /l(' U m '2)> 

and the number of independent variables is reduced to three. In 
this case the system is said to possess three degrees of freedom.* 
At temperatures over 135 138 the system contains N0. 2 alone, 
and behaves as a perfect gas with two degrees of freedom. 

In general, if a system contains m dependent and n independent 
variables, say 

x lt x. 2 , x s , . . . x n + m 

variables, the state of the system can be determined by m + n 
equations. As in the familiar condition for the solution of simul- 
taneous equations in algebra, n independent equations are required 
for finding the value of n unknown quantities. But the state of 
the system is defined by the m dependent variables ; the remaining 
n independent variables can therefore be determined from n inde- 
pendent equations. 

Let a given system with n degrees of freedom be subject to 
external forces 

**! -^2> ^8 ' ' ' ^* 

so that no energy enters or leaves the system except in the form 
of heat or work, and such that the n independent variables are 
displaced by amounts 

dx, dx lt dx. 2 , . . . dx n . 

Since the amount of work done on or by a system is measured by 
the product of the force and the displacement (page 182), these 

* If a system contains more than three degrees of freedom its state cannot be 
represented on a single diagram. 



g 51. COORDINATE OR ANALYTICAL GEOMETRY. 117 

external forces X-^X^ . . . perform a quantity of work dW which 
depends on the nature of the transformation. Hence 

dW = X^ + X> 2 dx., + . . . X n dx n 

where the coefficients X lt X 2 , X s . . . are called the generalised 
forces acting on the system. Duhem in his great work, Traite 
Elementaire de Mecanique Chimique fondee sur la Thermo - 
dynamique, 4 vols., 1897-99, makes considerable use of generalised 
forces and generalised coordinates. 



118 



CHAPTEE III. 
FUNCTIONS WITH SINGULAR PROPERTIES. 

52. Continuous and Discontinuous Functions. 

"Although a physical law may never admit of a perfectly abrupt 
change, there is no limit to the approach which it may make 
to abruptness." W. STANLEY JEVONS. 

THE law of continuity affirms that no change can take place 
abruptly. The conception involved will have been familiar to the 
reader from the second section of this work. It was there 
shown that the amount of substance (x) formed in a given time 
becomes smaller as the interval of time (t) during which the 
change occurs is diminished, until finally, when the interval of 
time approaches zero, the amount of substance formed also ap- 
proaches zero. In such a case x is not only a function of t, but 
it is a continuous function of / . 

The course of such a reaction may be represented by the motion 
of a point along the curve 

*-/(*). 

According to the principle of continuity, in order that the moving 

point may pass from one end (a) of the curve to the other (6), 
it must successively assume all values intermediate between 
a and b, and never move off the curve. This is a characteristic 
property of continuous functions. Several examples have been 
considered in preceding chapters. Most natural processes can be 
represented by continuous functions. Hence the old empiricism : 
Natura non agit per saltum. 

The law of continuity, though tacitly implied up to the present, 
is by no means always true. Even in some of the simplest phe- 
nomena exceptions may arise. In a general way, we can divide 
discontinuous functions into two classes : first, those in which the 
graph of the function suddenly stops to reappear in some other 



<:,;i. n NCTIONS wi in siN(;ri.\i; n:< >i'i;i; riKs. m 

part of the plane, in other words a " break " occurs ; second, 
those in which the graph suddenly changes its direction without 
exhibiting a break.* Other kinds of discontinuity may occur, but 
do not commonly arise in physical work. For example, a function 
is said to be discontinuous when the value of the function // = /(./) 
becomes infinite for some particular value of x. Such a dis- 
continuity occurs when x = in the expression y = 1 x. The 
differential coefficient of this expression, 

dx x' 2 ' 

is also discontinuous for x = 0. Other examples which may be 
verified by the reader are log x, when x = 0, tan x, when x = TT, . . . 
The graph for Boyle's equation, pv = constant, is also discontinuous 
at an infinite distance along either axis. 

53. Discontinuity accompanied by " Breaks ". 

The specific heat that is to say, the amount of heat required to 
raise the temperature of one gram of a solid substance one degree 
may be a known function of the temperature of the solid. As soon 
as the substance begins to melt, it absorbs a great amount of heat 
(latent heat), unaccompanied by any rise of temperature. When 
the substance has assumed the fluid state of aggregation the specific 
heat is again a function of the temperature until, at the boiling 
point, similar pheno- 
mena recur. Heat is ab- 
sorbed unaccompanied 
by any rise of tempera- 
ture (latent heat of 
vaporisation) until the 
liquid is completely 
vaporised. 

If the quantity of --, 

heat (Q) supplied be 
regarded as a function 
of the temperature (0), the curve (Fig. 55), represented by the 

equation 

y = <f>(x) ; or, Q = 



* Sometimes tlie word "break" is use<l indiscriminately for both kinds of dis- 
continuity. It is, indeed, questionable if ever the "break" is real. 



120 HIGHER MATHEMATICS. 54. 

is said to be discontinuous between the values Q = AB and CD, 
and breaks are said to occur in these positions. f(0) is therefore 
a discontinuous function, for, if a small quantity of heat be now 
added to the substance, the temperature does not change in a 
corresponding way. 

The geometrical signification of these phenomena is as follows : 
For the points A and B, corresponding to one abscissa, there are 
two, generally different, tangents to the curve, namely, tan a and 
tan a'. In other words (see page 82), 

-j0 = (f>'(6) = tan a = tan angle OH A ; 
and = <'(#) = tan a = tan angle BRA, 




that is to say, a function is discontinuous when the differential 
coefficient has two distinct values determined by the slope of the 
tangent to each curve at the point ivhere the discontinuity occurs. 

The physical meaning of the discontinuity in this example, is 
that the substance may have two values for its specific heat at the 
melting point, the one corresponding to the solid and the other to 
the liquid state of aggregation. The tangent of the angle repre- 
sented by the ratio dQ/dO obviously 
represents the specific heat of the sub- 
stance. An analogous set of changes 
occurs at the boiling point. 

Fig. 56 shows the result of plot- 
ting the variations in the volume of 
phosphorus with temperatures in the 
neighbourhood of its melting point. 
AB represents the expansion curve 
0^~ " of the solid, CD that of the liquid. 

FlG - 56> A break occurs between B and C. 

Phosphorus at its melting point may thus have two distinct 
coefficients of expansion, the one corresponding to the solid and 
the other to the liquid state of aggregation. 

54. The Existence of Hydrates in Solution. 

The fact (page 100) that an equation of the second (or nth) 
degree may include not only a single curve of the second (or nth) 
order, but also two (or n) straight lines, has been used in an in- 



FUNCTION- WITH SINGULAR PROPERTIES, 



genious way to indicate the probable existence of certain chemical 
compounds in solution. The following data are quoted at some 
length in order to explain an important application of mathematical 
methods for bringing these obscured lines into prominence : 

If p denotes the percentage compositions of various aqueous 
solutions of ethyl alcohol and s the corresponding specific gravities 
in vacua at 15 (sp. gr. H 2 at 15 = 9991-6), we have the follow- 
ing table compiled by Mendel6eff : 



P 


s 


P 


s 


P * 


p 


5 
10 

15 
20 
25 


9904-1 
9831-2 
9768-4 
9707-9 
9644-3 


30 
35 
40 
45 
50 


9570-2 
9484-5 
9389-6 

9287-8 
9179-0 


55 ! 9067-4 
60 8953-8 
65 8838-6 
70 8714-5 
75 8601-4 


80 8479-8 
85 8354-8 
90 8225-0 
95 8086-9 
100 7936-6 
1 



It is found empirically that the experimental results are fairly 
well represented by the equation 

s = a + bp + cp-, ... (1) 

which is the general expression for a parabolic curve, a, b and c 
being constants (page 77). By plotting the experimental data the 
curve shown in Fig. 57 is obtained. 


























**l 


^ 








j 

! 














"^S 


^ 


^ l 


















^s^ 










. 








x 


>> 

























I 





* 

] 


ft 
TIG. 


6 

57. 


O J9O 


10 



It is urged that just as compounds may be formed and decom- 
posed at temperatures higher than that at which their dissociation 
commences, and that for any given temperature a definite relation 
exists between the relative amounts of the original compound 
and of the products of its dissociation, so may solutions contain 
definite but unstable hydrates at a temperature above their dis- 



122 



HIGHER MATHEMATICS. 



$54. 



sociation temperature. If the dissolved substance really enters 
into combination with the solvent to form different compounds 
according to the nature of the solution, many of the physical 
properties of the solution (density, thermal conductivity and such 
like) will naturally depend on the amount and nature of these 
compounds, because chemical combination is usually accompanied 
by volume, density, thermal and other changes. Assuming that 
the amount of such a definite compound is proportional to the 
concentration of the solution, the Tate of change of, say, the 
density with change of concentration will be a linear function 
of p, that is to say, from the differentiation of (1) 

ds 

%-* + **, < 2 > 

W 7 here ds is the difference in the density of two experimental 
values corresponding to the difference in the percentage com- 
position of two solutions of the same substance.* 

The second member of (2) corresponds with the equation of a 
straight line (page 69). On treating the experimental data by 
this method, Mendeleeff f found that dsjdp was discontinuous, and 

. ___ that breaks 

,0 l!2H 2 0| , . . , 

were obtained 

by plotting dsfdp 
as ordi nates 
against abscissa 
p for concen- 
trations corres- 
ponding to 17 '56, 
FIG. 58 (after Mendeleeff). Q.QQ and 88'46 

per cent, of ethyl alcohol. These concentrations coincide with 
chemical compounds having the composition C.,H b OH . 12H.,0, 
C. 2 H b OH . 3H. 2 and SC^H^OH . H 2 as shown in Fig. 58. 

This procedure has been extensively used by Pickering | in the 
treatment of an elaborate and painstaking series of determinations 
of the physical properties of solutions. 

Crompton found that if the electrical conductivity of a solution 

* See page 247 for the method of finding dyjdx from a set of tabulated measure- 
ments. 

, f Mendeleeff, Journal oft/ie London Chemical Society, 51, .778, 1887. 

J Pickering, Journal London Chemical Society and Philosophical Magazine, about 
1890. Crompton, Journal Chemical Society, 53, 116, 1888. 




< :,4. FUNCTIONS \VIIH BINGULAB Nioi'Ki; 1 1 i->. IL-.; 

is regarded as a function of its percentage composition, such that 
K = a f bp + cp- + fp\ ... (3) 

the rirst differential coefficient gives a parabolic curve of the type 
of (1) above, while the second differential coefficient, instead of 
being a continuous function of p, 



was found to consist of a series of straight lines, the position of the 
breaks being identical with those obtained by Mendeleeff for the 
first differential coefficient ds/dp. The values of the constants A 
and B are obvious. 

The mathematical argument is that the differential coefficient 
of a continuous curve will differentiate into a straight line or 
another continuous curve ; while if a curve is really discontinuous, 
or made up of a number of different curves, it will yield a series of 
straight lines. Each line represents the rate of change of the 
particular physical property under investigation with the amount 
of hypothetical unstable compound existing in solution at that 
concentration. An abrupt change in the direction of the curve 
leads to a breaking up of the first differential coefficient of that 
curve into two curves which do not meet. For the p, s-curve, 
dsjdp is discontinuous ; for the ds/dp, p-curve, d-s, dp' 2 is dis- 
continuous. 

It must be pointed out that the differentiation of experimental 
results very often furnishes quantities of the same order of 
magnitude as the experimental errors themselves.* This is a 
very serious objection. Pickering has proposed to eliminate the 
experimental errors to some extent by differentiating the results 
obtained by "smoothing" the curve obtained by plotting the 
experimental results. f On the face of it this "smoothing"* of 

* This will appear after reading Chapter V., 104. 

fSee Horstmann (Liebig's Annalen, Suppl., 8, 125, 1872) for finding dpfilQ by 
drawing tangents to the graph of the experimental data; and Berichte, 2, 137, 1869, 
for finding ftp/dd by the differentiation of an empirical equation. See 104. 

^ The results of the observation of a series of corresponding changes in two 
variables are plotted as abscissae and ordinates by light dots on a sheet of squared 
paper, and a curve is drawn to pass as nearly as possible through all these points. 
The resulting curve is assumed to be a graphic representation of the general formula 
(known or unknown) connecting the results of experiment. Points deviating from the 
curve are assumed to be due to errors of observation. As a general rule the curve 
with the least curvature is chosen to pass through or within a short distance of the 



124 HIGHER MATHEMATICS. 55. 

experimental results is a dangerous operation even in the hands of 
the most experienced workers. Indeed, it is supposed that that 
prince of experimenters, Eegnault, overlooked an important pheno- 
menon in applying this very smoothing process to his observations 
on the vapour pressure of saturated steam. Eegnault supposed 
that the curve showed no singular poinfc when water passed from 
the liquid to the solid state at 0. It was reserved for J. Thomson 
to prove that the ice-steam curve is really different from the water- 
steam curve (see page 127). 

55. Discontinuity accompanied by Change of Direction. 

The vapour pressure of a solid increases continuously with 
rising temperature, until at its melting point the vapour pressure 
suddenly changes. This is shown graphically in Fig. 59. The 
point P marks the melting point of the substance. The curve 
does not exhibit a break because the vapour 
pressure is the same at this point whether 
the substance be solid or liquid. 

It is, however, quite clear that the tan- 
gents of the two curves differ from each 
other at the transition point P, because 

... .. _ ; , dv dp 

tjp~ ,R' tana =/(0) = ^ and, tan a' = /(0) = jy 

FlG - 59. if the equations to the two curves were 

ax + by = 1 and bx + ay = l, the roots of the equations x = I/ (a + b) 
and y = l/(a + b) would represent the coordinates of the point of 
intersection (see page 73). 

To illustrate this kind of discontinuity we shall examine the 
following phenomena : 

(1) Critical temperature. Cailletet and Collardeau have an 
ingenious method for finding the critical temperature of a 
substance without seeing the liquid.* By plotting temperatures 

greatest number of dots, so that an equal number of these dots (representing ex- 
perimental observations) lies on each side of the curve. Such a curve is said to be a 
smoothed curve. The choice of the proper curve is more or less arbitrary. Pickering 
used a bent spring or steel lath held near its ends. Such a lath is shown in statical 
works to give a line of constant curvature. E.g., Minchin's A Treatise on Stotics, 2, 
306, 1886. 

* Cailletet and Collardeau, Ann. (U'Vhini. et de Phys. [6], 25, 522, 1891. Note 
that the critical temperature is the temperature above which a substance cannot exist 
other than in the gaseous state. 




FUNCTIONS WITH 9MGULAB PROPERTIES. 



i _:, 




FIG. 60. 



ibscissae against the vapour pressures of different weights of 
the same substance heated at constant volume, a series <.t curves 
are obtained which are coincident as lon.n as part of the substance 
is liquid, for " the pressure exerted by a 
saturated vapour depends on temperature 
only and is independent of the quantity of 
liquid with which it is in contact ". Above 
the critical temperature the different masses 
of the substance occupying the same vol- 
ume give different pressures. From this 
point upwards the pressure-temperature 
curves are no longer superposable. A 
series of curves are thus obtained which 
coincide at a certain point P (Fig. 60), the abscissa of which 
denotes the critical temperature. As before, the tangent of each 
curve Pa, Pb ... is different from that of OP. 

(2) Coexistence of the different states of aggregation. 
Another example which is also a good illustration of the beauty 
and comprehensive nature of the graphic method of representing 
natural processes may be given here. 

(a) When water, partly liquid, partly vapour, is enclosed in a 
vessel, the relation between the pressure and the temperature can 
be represented by a curve PQ (Fig. 61), 

which gives the pressure corresponding 
to any given temperature when the liquid 
and vapour are in contact and in equi- 
librium. This curve is called the steam 
line. 

(b) In the same way if the enclosure 
were filled with solid (ice) and liquid 
water the pressure of the mixture would ot 

be completely determined by the tern- F IG. 61. Triple Point, 
perature. The relation between pressure and temperature is 
represented by the curve NP, called the ice line. 

(c) Ice may be in stable equilibrium with its vapour, and we 
can plot the variation of the vapour pressure of ice with its tem- 
perature. The curve PM so obtained represents the variation of 
the vapour pressure of ice with temperature. It is called the hoar 
frost line. 

The plane of the paper is thus divided into three parts bounded 




126 HIGHER MATHEMATIC8. 55. 

by the three curves PM, PN, PQ. If a point falls within one of 
these three parts of the plane, it represents some state in which 
the water may exist in the form of ice, liquid or steam as the case 
might be.* If the point falls on a boundary line it corresponds to 
the coexistence of two states of aggregation. Finally, at the point 
P, and only at this point, the three states of aggregation, ice, water 
and steam may coexist together. This point is called the triple 
point. For water the coordinates of the triple point are 

p = 4-57 mm., 6 = 0-00747 C. 
The two formulae, 

dQ = Od^; (*Qftv) e = OQpftB)., 

were discussed in one of the examples appended to 26. Divide 
the former .by dv and substitute the result in the latter. We thus 
obtain, 





which states that the change of entropy (</>) per unit change of 
volume (v), at constant temperature (0 absolute), is equal to the 
change of pressure per unit change of temperature at constant 
volume. 

If a small amount of heat (dQ) be added to a substance existing 
partly in one state, 1, and partly in another state, 2, a proportional 
quantity (dm) of the mass changes its state, such that 

dQ = L u dm t 

where Z/ 12 is a constant representing the latent heat of the change 
from state 1 to state 2. By definition of entropy (</>), 

dQ = Od<f> ; hence d$ = -^dm . . (2) 

If v lt v 2 be the specific volumes of the substance in the first 
and second states respectively 

dv = v^drn - v^dm = (v 2 - v-^dm. 
From (2) and (1) 

\ A* . (*P\ _ L u ,ox 

)o~ O^-v,)' \DO) 9 0(v, - v,)' 

This last equation tells us at once how a change of pressure 
will change the temperature at which two states of a substance 
can coexist provided that we know v v v. 2 , 6 and L 12 . 

* Certain unstable conditions (metastable states) are known in which a liquid may 
be found in the solid region. A supercooled liquid, for instance, may continue the QP 
curve along to S' instead of changing its direction along PM. 



FUNCTION'S \virn BINOULAB PROPERTIES, H'7 

:PLES. (1) If the specific volume of ice is 1-087, and that of water 
unity, find the lowering of the freezing point of water when the pressure 
increases one atmosphere (latent heat of ice = HO cal.). Here r a - r, = 0-87, 
= 273, dp = 7G cm. mercury. The specific gravity of mercury is 13'5, and 
the weight of a column of mercury of one square cm. cross section is 
76 x 13-5 = 1,033 grams. Hence dp = 1,033 grams, L 12 = 80 cal. = 80 x 47,600 
( '.<,.S. or dynamical units. From (3), do = 0-0072 C. per atmosphere. 

or naphthalene 6 = 352-2, i: 2 - v l = 0-146; L n - 35-46 cal. Find 
the change of melting point per atmosphere increase of pressure. dB = 0*035. 

Let L 10 , L 23 , L ai be the latent heats of conversion of a substance 
from states 1 to 2, 2 to 3, 3 to 1 respectively ; r lf v 2 , v z the re- 
spective volumes of the substance in states 1, 2, 3 respectively ; 
let denote the absolute temperature at the triple point. Then 
dp.JO is the slope of the tangent to these curves at the triple 
point, and 



The specific volumes and the latent heats are generally quite 
different for the three changes of state, and therefore the slopes of 
the three curves at the triple point are also different. 

The difference in the slopes of the tangents of the solid- vapour 
(hoar frost line) and liquid-vapour (steam line) curves of water 
(Fig. 59) is 



-i * 8 - 

At the triple point L 13 = L V2 + L. 2Z , and (r 3 - i^) = (v. 2 - r t ) + (r 3 - r._,). 

EXAMPLE. As a general rule, the change of volume on melting, (r a - t^), 
is very small compared with the change in volume on evaporation, (r. - r 2 ), 
or sublimation, (r, - v^ ; hence r., - i\ may be neglected in comparison with 
the other volume changes. Then, 

/p\ _ (dp\ L 

\-deJ 15 \-d9j n ~ e(r,- r,)' 

Hence calculate the difference in the slope of the hoar frost and steam lines 
for water at the triple point. Latent heat of water = 80 ; I/ 12 = 80 x 42,700 ; 
B = 273, ?-, - r. 2 = 209,400 c.c. Substitute these values on the right-hand side 
of the last equation. Ansr. - 059. 

The above deductions have been tested experimentally in the 
case of water, sulphur and phosphorus; the results are in close 
agreement with theory. A full discussion of the properties of 
sulphur, water and phosphorus, etc., in relation to the triple point, 
are given by Duhem in his Traite Elementaire de Me'caniqm 



V 



128 



HIGHER MATHEMATICS. 



Chimique fondee sur la Thermodynamique, 2, 93 ; an outline 
sketch will also be found in Preston's Theory of Heat (1894), 
pp. 677-8. 

(3) Cooling curves. If the temperature of cooling of pure 
liquid bismuth be plotted against time, the resulting curve will 
be continuous (ab, Fig. 62), but the moment a part of the metal 
solidifies, the curve will take another direction be, and continue 

so until all the metal is solidified, 
when the direction of the curve 
again changes, and then continues 
quite regular along cd. For bis- 
muth the point b is at 268. 

If the cooling curve of an alloy 
of bismuth, lead and tin (Bi, 21 ; 
Pb, 5*5 ; Sn, 75*5) is similarly plot- 
ted, the first change of direction is 
observed at 175, when solid bis- 
muth is deposited; at 125 the curve 
again changes its direction, with a 




FIG. 62. Cooling Curves. 



simultaneous deposition of solid bismuth and tin ; and finally at 
96 another change occurs corresponding to the solidification of 
the eutectic alloy of these three metals. 

These cooling curves are of great importance in investigations 
on the constitution of metals and alloys. The cooling curve of 
iron from a white heat is particularly interesting, and has given 

rise to much discussion. The 
curve shows changes of direction 
at about 1,130, at about 850 
(called Ar% critical point), at 
about 770 (called Ar. 2 critical 
point), at about 500 (called the 
At\ critical point), at about 450 
500 C., and at about 400 C. 
(below redness). The magnitude 
*?* of these changes varies according 
FIG. "63. Portion of Cooling Curve of to the purity of the iron. Some 
Iron - are very marked even with the 

purest iron. This sudden evolution of heat (recalescence) at 
different points of the cooling curve has led many to believe 
that iron exists in some allotropic state in the neighbourhood of 




FUNCTIONS WITH siN<;ri.AR PROI'KK I 1 1.- 



129 



these temperatures.* Fig. 63 shows part of a cooling curve of 
iron in the most interesting region, namely, the Ar z and Ar 2 
critical points. 

56. Maximum and Minimum Values of a Function. 

If a mixture of hydrogen and chlorine gases is exposed to a 
ray of light, the amount of chemical action which takes place in 
a given time depends on the wave length of the light, that is to 
say, if y denotes the amount of hydrogen chloride formed in unit 
time, and x the wave length of light, y = f(x). Experiment shows 
that as x changes from one value to another, y changes in such a 
way that it is sometimes increasing and sometimes decreasing. 
In consequence, there must be certain values of the function for 
which y, which had 
previously been in- 
creasing, begins to 
decrease, that is to 
say, y is greater for 
this particular value 
of x than for any 
adjacent value ; in 
this case y is said 
to have a maxi- 
mum value. Con- 
versely, there must 
be certain values of 
f(x) for which y, 
havin<* been de- Wwv ten^/i of aclinic rays referred hFrawiko^ lines. 
creasing, begins to FlG ' 64 (Diagrammatic), 

increase. When the value of y, for some particular value of x, is 
less than for any adjacent value of x, y is said to be a minimum 
value. 

Fig. 64 is a geometrical illustration of the action of light rays 
of different wave length on a mixture of hydrogen and chlorine. 
Imagine the variable ordinate of the curve to move perpendicularly 
along Ox, gradually increasing until it arrives at the position PM , 

* Roberts- Austen' s papers in the Proceedings of the Society of Mechanical En- 
gineers for 1891, 543 ; 1893, 102 ; 1895, 238 ; 1897, 31 ; 1899, 35, may be consulted for 
fuller details. 

I 




130 HIGHER MATHEMATICS. 57. 

and afterwards gradually decreasing. The ordinate at PM is said 
to have a maximum value. The decreasing ordinate, continuing its 
motion, arrives at the position QN, and after that gradually in- 
creases. In this case the ordinate at QN is said to have a minimum 
value. 

The terms " maximum " and "minimum" do not necessarily 
denote the greatest and least possible values which the function 
can assume, for the same function may have several maximum and 
several minimum values, any particular one of which may be greater 
or less than another value of the same function. 

EXAMPLE. If the ?/-axis represents the amount of hydrogen chloride 
formed in unit time ; the aj-axis, the wave length of the ray of light impinging 
on a mixture of hydrogen and chlorine gases, interpret the curve shown in 
Fig. 64. 

The mathematical form of the function employed in the above 

illustration is unknown, the curve is an approximate representation 

of corresponding values of the two variables determined by actual 

* measurements. (Bunsen and Eoscoe, Phil. Trans., 148, 879, 1859.) 

EXAMPLE. Plot the curve represented by the equation 

y - since. 
Give x a series of values ^ir, TT, -|TT, 2ir, and so on. 

Maximum values of y occur for x = far, fir, f?r, . . . 
Minimum values of y occur f or x = - ^TT, f *, TT, . . . 
The resulting curve is an harmonic or sine curve (see Fig. 51, page 112). 

One of the most useful applications of the differential calculus is 
the determination of maximum and minimum values of a function. 
Many of the following examples can be solved by special algebraic 
or geometric devices. The calculus, however, offers a sure and 
easy method for the solution of these problems. 



57. How to find Maximum and Minimum Values of a 

Function. 

Let us trace the different values which the tangent to the 
curve shown in Fig. 65 may assume. Firstly, when x is increasing, 
y is approaching a maximum value and the tangent to the curve 
makes an acute angle with the a?-axis. In this case, Table XII I., 

dy . 
tan a and . . T- is positive ; 



FUNCTIONS WITH SINGULAR PBOPEBTII->. 



at P the tangent is parallel to the rr-axis, that is to say, 

dy 

tan a and also , are zero . . (1) 

ax 

Secondly, immediately after passing P, the tangent to the curve 
makes an obtuse angle with the #-axis, that is- to say, 

dy 
tan a and -, are negative . . (2) 

Finally, as the tangent to the curve approaches the minimum value 
QN, dy/dx remains negative ; at Q the tangent is again parallel to 
ic-axis, an<P 



dy 
tan a, as well as -,- , is zero. . 



(3) 



After passing Q, dy/dx again becomes positive. 

There are some curves which have maximum and minimum 
values very much resembling P and Q' (Fig. 66). These curves 
are said to have cusps at P' and Q'. 

u ' 





FIG. 65. Maximum and Minimum. 



M' 



FIG. 66. Maximum and 
Minimum Cusps. 



It will be here observed that x increases and y approaches a 
maximum value while the tangent P'M' makes an acute angle with 
the x-axis, that is to say, dy/dx is positive. At P' the tangent 
becomes perpendicular to the x-axis, and in consequence the ratio 
dy/dx becomes infinite. After passing P', dy/dx is negative. In 
the same way it can be shown that as the tangent approaches Q'N', 
dy/dx is negative, at Q', dy/dx becomes infinite, and after passing 
Q', dy/dx is positive. 

We thus deduce the following rules : 

(1) When the first differential coefficient changes its sign from a 
positive to a negative value the function has a maximum value, and 
when the first differential coefficient changes its sign from a negative 
to a positive value the function has a minimum value. 






132 HIGHER MATHEMATICS. $ 58. 

(2) Since a function can only change its sign by becoming zero 
or infinity, it is necessary for the first differential coefficient of the 
function to assume either of these values in order that it may have 
a maximum or a minimum value. 

(3) In order to find all the values of x for which y possesses a 
maximum or a minimum value, the first differential coefficient must 
be equated to zero or infinity and the value of x which satisfies these 
conditions determined. 

EXAMPLES. (1) Consider the equation y = x 2 - 8x, 

' ' dx 

Equating the first differential coefficient to zero, we have 

2x - 8 = ; or x = 4. 

Add + 1 to this root and substitute for x in the original equation, 
when x = 3,y= 9 - 24 = - 15 ; 
x = 4, y = 16 - 32 = - 16 ; 
x = 5, y = 25 - 40 = -' 15. 

y is therefore a minimum when x = 4, since a slightly greater or a slightly 
less value of x makes y assume a greater value. If the values of y had been 
less f or x = 3 and x 5, than for x = 4, then, x = 4 would have made y a 
maximum. If one had been greater, and the other less than for x - 4, this 
root would have been neither a maximum nor a minimum. 

The addition of + 1 to the root gives only a first approximation, as will 
be shown later on (page 392). The minimum value of the function might, for 
all we can tell to the contrary, lie between 3 and 4 or 4 and 5. The approxi- 
mation may be carried as close as we please by using less and less numerical 
values in the above substitution. Suppose we substitute in place of + 1, + 8x, 
then 

when x = 4 - Sx, y = 8x 2 - 16 ; 
x = 4 ,y= - 16 ; 
x = 4 + 8x, y = 8;r 2 - 16. 

Therefore, however small 8x may be, the corresponding value of y is greater 
than - 16. That is to say, x = 4 makes the function a minimum. 
(2) Show that y = 1 + 8x - 2x 2 , has a maximum value for x = 2. 

58. Points of Inflection. 

Continuing the discussion in the preceding paragraph, to equate 

dy dy 

ax ax 

is not a sufficient condition to establish the existence of maximum 
and minimum values of a function, although it is a rough practical 
test. Some of the values thus obtained do not necessarily make 
the function a maximum or a minimum, since a variable may 



FUNCTIONS WITH SINGULAR PROPERTIES. 



188 



become zero or infinite without changing its sign. This is obvious 
from a simple inspection of Fig. 67, where 

( '- f = 0, or oo, resp., 
dx 

for the points R and S. Yet neither maximum nor minimum values 
of the function exist. A further test is therefore required in order 
to decide whether individual values of x correspond to maximum or 
minimum values of the function. This is all the more essential in 
practical work where the function, not the curve, is to be operated 
upon. 



r 



FIG. 67. Points of Inflection. FIG. 68. Concavity and Convexity. 

By reference to Figs. 67 and 68 it will be noticed that the 
tangent crosses the curve at the points R and S. Such a point is 
called a point of inflection. The point of inflection (or inflexion) 
marks the spot where the curve passes from a convex to a con- 
cave, or from a concave to a convex configuration with regard to 
one of the coordinate axes. The terms concave and convex have 
here their ordinary meaning. 

59. How to find whether a Curve is Concave or Convex 
with respect to the x-Axis. 

Referring to Fig. 68, along the convex part from A to B, the 
numerical value of tan a, regularly decreases to zero. At B the 
lowest point of the curve tan a = ; from this point to E the 
tangent to the angle continually increases, for tan a has now an 
increasing positive value. 

The differential coefficient of tana with respect to x for the 
convex curve ABU is 



</(tana) _ dfy 
dx. ~^ >U ' 



(1) 



134 HIGHER MATHEMATICS. 60. 

because, if a function, y = f(x), increases with increasing values of 
x, dy/dx is positive ; while if the function, y = f(x), decreases with 
increasing values of x, dy/dx is negative. Along the concave part 
of the curve RCS, tan a regularly decreases in value ; from R to C, 
tan a has a decreasing positive value. At the point C, tan a = 0, 
and from C to S, tan a has a continually increasing negative 
value. 

The differential coefficient of tan a with respect to the concave 
curve RCS is 



_ 
dx Sx* <{) ' 

Hence a curve is concave or convex with respect to the upper side 
of the x-axis, according as the second differential coefficient is 
-positive or negative. 

I have assumed that the curve is on the positive side of the r-axis ; when 
the curve lies on the negative side, assume the x-axis to be displaced parallel 
with itself until the above condition is attained. A more general rule, which 
evades the above limitation, is proved in the regular text-books. The proof is 
of little importance for our purpose. The rule is to the effect that " a curve is 
concave or convex with respect to the ic-axis according as the product of the 
ordinate of the curve and the second differential coefficient, i.e., according 

d z y 
as 2/3~2 ' 1S positive or negative ". 

EXAMPLES. (1) Show that the curves y = log a; and y = xlogx are re- 
spectively concave and convex towards the .r-axis. 

(2) Show that the parabola is concave upwards below the x--axis (where y 
is negative) and convex upwards above the #-axis. 

60. How to find Points of Inflection. 

From the above principles of curvature and points of inflection, 
it is clearly necessary, in order to locate a point of inflection, to 
find a value of x, for which tan a assumes a maximum or a minimum 

value. But 

dy 

tana = j-, 
ax 



' ' 

Hence the rule : In order to find a point of inflection at which 
the second differential coefficient changes sign, we must equate the 
second differential coefficient of the function to zero and find the 
value of x satisfying these conditions. 



- 



61. FUNCTIONS WITH SINGULAR PROPERTIES. 136 

EXAMPLES. (1) Show that the curve 

y = a + (x - 6) 3 

has a point of inflection at the point y = a, x b. Differentiating twice we 
get cPyjdx* = 6(x - b). Equating this to zero we get x = b, and hence sub- 
stituting in the original equation y = a. When x < b the second differential 
coefficient is negative, when x > 6 the second differential coefficient is posi- 
tive. Hence there is an inflection at the point (b, a). 

(2) For the special case of the harmonic curve 

dhj 
y = Bmx,fap = - sinx= - y, 

that is to say, at the point of inflection the ordinate y changes sign. This 
occurs when the curve crosses the it-axis, and there are an infinite number of 
points of inflection for which y = 0. 

(3) Show that the probability curve, y = ke h ***, has a point of inflection 



61. Multiple Points. 

A multiple point is one through which two or more branches of a curve 
meet or intersect. There are two species : 

(1) Two or more branches of the curve intersect. 

(2) Two or more branches of the curve meet but do not intersect (point of 
osculation). 

An algebraic equation of the nth degree has n roots corresponding to the 
different values of one of the variables. When two or more branches of a 
curve touch each other, the different values of y, corresponding to .r, become 
equal to each other, while for slightly less values of x, the corresponding values 
of y are not equal. 

First species of multiple point. If the first differential coefficient has two 
or more real values, the curve has more than one tangent, that is to say, the 
curves intersect. The number of intersecting branches is denoted by the 
number of real roots of the first differential coefficient. 

EXAMPLE. In the lemniscate curve, familiar to students of crystallography, 

y* = a*x 2 - x 4 ; y = x \/a 2 - x 2 . 

Here y has two values of opposite sign for every value of x between + a ; the 
curve is therefore symmetrical with respect to the avaxis. When .r = + a, 
these two values of y become zero ; but these are not multiple points since 
the curve does not extend beyond these limits, and therefore cannot satisfy 
the above conditions. When x = the two values of y become zero, and 
since there are two values of y one on each side of the point x = 0, y = 0, this 
is a multiple point. Since 



_ 

dx~~ v/a 2 - x* 

becomes + a when x 0, it follows that there are two tangents to the curve 
at this point, such that 

tan a = + a. 
Fig. 69 shows* this curve. 



136 



HIGHER MATHEMATICS. 



62. 



Second species of multiple point point of osculation. If the first differ- 
ential coefficient of a multiple point has two or more real and equal roots, 
the different, branches of the curve have a common tangent, and the point of 
contact is called a point of osculation. 




FIG. 69. Multiple Point (O). 




F IGL 70. Point of Osculation (P). 



EXAMPLE. In the curve y - (x - 1) (x - 2) (x - 3), for values of x other 
than and - 1 there are at least two values of y ; dyjdx = 3x 2 - 12x + 11 
vanishes when x = 2 + 1/*J3; hence the two branches of the curve are 
tangents to each other at this point, which is therefore a point of osculation. 
The curve is shown in Fig. 70. 



62. Cusps. 

A cusp is a point where two branches of a curve have a common tangent 
and stop at that point, There are two species : 

(1) The two branches lie on opposite sides of the common tangent. 

(2) The two branches lie on the same side of the common tangent. 

The cusp is therefore a special case of the point of osculation, where the 
branches terminate at the point of contact instead of passing beyond. Hence 
the values of y on one side of the point are real and on the other, imaginary.* 

To distinguish cusps from points of osculation : compare the ordinate of 
the curve for that point with the ordinates of the curve on each side. For a 
cusp, y and the first differential coefficient have only one real value. 

First, cusps of the first species (or " keratoid cusps ") have two values for 
the second differential coefficient differing only in sign. The meaning of this 
will be clear from pages 133 to 134. 

EXAMPLE. In the cissoid curve, y = b + v (x 2 - a 2 ) 3 , y is imaginary for 
all values of x between + a. When x = a, y has one value ; for any point to 

right of x=+a or to the left of x= - a, y has two values dyjdx= Sx(x 2 - a?) 
vanishes when x = a. The two branches of the curve have therefore a com- 
mon tangent parallel to the x-axis and there is a cusp. Next determine 
d?yldx* = f (x* - a 2 ) ~ * 



* For imaginary quantities read footnote, page 175. 



64. FUNCTIONS WITH SINGULAR PROPERTIES. 



137 



and substitute some value for a, say a + h. We then find that the cusp is 
of the first species with the upper branch + f (2ah + /t 8 ) ~ , convex towards 

the x-axis ; and the lower branch - $(2a/t -f h z ) ~ , concave towards the x-axis. 
The curve is shown in Fig. 71. 

Second, cusps of the second species (or "rhamphoid cusps") have two 
different values for the second differential coefficient of the same sign. 





FIG. 71. Cusps of the First Species. 



FIG. 72. Cusp of the 
Second Species. 



EXAMPLE Show that the curve (y - a; 2 ) 2 = x 5 has a cusp of the second 
species at the origin. The lower curve also has a maximum when x = . 
The general form of the curve is that shown in Fig. 72. 

It will perhaps amuse the reader to investigate the properties of the 
following curves : 

r = a sin 26 ; r = a sin 66 ; 
r 3 = 3 cos 4 f 6 ; r* = a 5 cos 5 f 0. 

63. Conjugate or Isolated Point. 

A conjugate point, or acnode, is one whose coordinates satisfy the equation 
to the curve, and yet is itself detached from the curve. 

If a point is isolated from every part of the curve, it follows that on each 
side of this point real values of one coordinate must give a pair of imaginary 
values of the other. This may be determined by successive substitution of 
x + Sx, x - 5x, etc. 

EXAMPLE. Show that the origin in the graph of ay z = x 2 (x - b) is a 
conjugate point. 

When one branch of a curve suddenly stops we have a point d'arret or 
terminal point (see Fig. 120). 

EXAMPLE. The origin in the two transcendental curves y = a 1 /*, where 
a is greater than unity and y = x log a. 



64. Asymptotes. 

As explained on page 87, an asymptote is a straight line which approaches 
closer and closer to a given curve, as x or y increases without limit. It is often 
defined as the limiting position of a tangent to a curve when the point of 
contact moves an infinite distance away (see the lines OP, OF, Fig. 28; Oc, 
Fig. 29 ; Op, Fig. 125, etc.). 



138 HIGHER MATHEMATICS. 64. 

Let OPS (Fig. 73) be a plane curve, BP a tangent to the curve at the 
point P(x l , T/J). If BP intersects the ^/-axis at the point (0, y), and the cc-axis 
at the point (x, 0), then (6), 38, 

y - 1/1 = fo& - *i)- 

If, we put y - 0, the intercept of the tangent with the cc-axis is 

x = OB = x, - 7/1^, (2) 

and if x = we get the intercept of the tangent with the ?/-axis, 

T 



If, when x l or y l becomes infinite, either x or y is finite, the curve will have 

one or more asymptotes which can be de- 
termined. The following deductions may 
be made : 

(1) If when x = oo, y is finite, the 
asymptote is parallel to the ic-axis. 

(2) If when x is finite, y = CD, the 
asymptote is parallel to the 7/-axis. 

(3) If x and y are both finite, the 
asymptote passes through (0, y) and (x, 0). 

(4) If x and y are both zero, the asymp- 
FIG. 73. ^Q^e passes through the origin, and its 

direction is determined by the value of yjx when x or y is infinite. 

(5) If x and y are both infinite, the tangent is at an infinite distance from 
the origin, and cannot be constructed since it is indeterminate. 

EXAMPLES. (1) Determine whether the hyperbola has asymptotes. The 
hyperbola 




has two real values of y, however great x may be, and hence the curve has two 
infinite branches to the right. Differentiating the above equation 

dx _ a 2 y 2 __ x 2 - a 2 
y dy ~ b 2 ' x~ ~ x 

a 2 b 2 

if x is infinite, OB = a' 2 /^ = ; and if y is infinite O(0, y) = - b 2 ^ = 0, 
that is to say, the hyperbola has two asymptotes passing through the origin as 
(4) above. The direction of- the asymptotes is obtained by putting 

dy b*x __ bx 

dx~ a?y ~ a~J(x 2 - a 2 )' 

when x = + oo, dy/dx = + bja. Hence the asymptotes are the produced diagonals 
of a rectangle described on the axes. If x - oo, there is another pair of 
infinite branches having the same lines through the origin as asymptotes. 

(2) Has the parabola y 2 ax an asymptote ? No. 0(x, 0) = - x, 
O(0, y) = %y. When x is infinite, O(x, 0) = - CD, and when y is infinite 
0(0, y) = + oo. Hence the parabola has no asymptotes as in (5) above. 

(3) Show that the logarithmic curve x = log y (and also y = e x ) has an 
asymptote coincident with the abscissa axis, and a branch of the curve ex- 
tending to the right, not asymptotic (case (5) above). 



WITH sINdl -\.\\i 
65. Summary. 

(1) Equating the first differential coefficient of a function to zero gives a 

dy 
maximum 01 minimum value. If the sign of ^ changes from + to -- when x is 

substituted, ij is a maximum ; if the sign changes from - to + , y is a minimum ; 

<l-i/ 
also, if ^. 2 is positive, y is a maximum, if negative, a minimum (see 102). 

d?t( 

(2) If V-TJ is positive, the curve is concave towards the x'-axis, if negative 

VI rtlt/ 

OOfKMe. 

(3) If ~T 0, but does not change sign when x is substituted, we have a 

d?t/ 
[>oint of inflection for which -r-^ = 0. 

di/ 

(4) If -T- = oo and changes its sign, there is a cusp, which is a maximum or 

a minimum according to its sign. 

(5) If -^7 = oo and y = oo without changing sign, y is an asymptote. 



(6) If = and 1 J = wit h a change of sign, y has an infinite maxi- 
mum value. 

(7) If V~ has two or more unequal values, a multiple point occurs. 

(8) If 3^7 has two or more equal values, a point of osculation occurs. 

(9) If -5- and y have one real value, and the value of y on one side of the 

d?y 
point is imaginary, we have a cusp : of 1st species, if ^-3 has two values, differing 



only in sign ; of 2nd species, if -, r . 2 has two different values, of the same sign. 

(10) If ^ and hig 
have a conjugate point. 



(10) If and higher differential coefficients have impossible values, we 



66. Curvature. 

The curvature at any point of a plane curve is the rate at 
which the curve is bending. Of two curves AC, AD, that has 
the greater curvature which departs the __ ^A _ 
more rapidly from its tangent AB (Fig. ^^^ 
74). The angle between the tangents at ' ( ^D 

the ends of an arc of the curve is called FlG - 74 - 

the total curvature of the arc. In passing from any point P 
(Fig. 75) to another neighbouring point P l along any arc 8s of 
the plane curve AB, the tangent at P turns through the angle oa 



140 HIGHER MATHEMATICS. g 66. 

where a is the angle made by the intersection of the tangent at P 
with the #-axis. The angle 8a is the total curvature of the arc 
under consideration, and the ratio 
(total curvature) So. 
(length of arc) = Ss = ( mean cu ature of arc). 

The curve turns through the angle Sa in the lengthJSs, and therefore 
the total curvature is the limiting value of 

Lt&a/Ss = da/ds = (rate of bending of curve) . (I) 

The curvature of the circumference of all circles of equal radius 

is the same at all points, and varies inversely as the radius. This 

is established in the following way : In the circle (Fig. 76), is 

the centre, r, r are radii. From elementary geometry, the angle 





FIG. 75. FIG. 76. 

RSQ = angle POQ. The angle POQ is measured in circular 

measure (page 494) by the ratio of the arc PQ to the radius, i.e., 

angle POQ = arc PQ/r, or Sa/Ss = 1/r, 

da 1 

/. (curvature of circle) = -r- = - - . (2) 
as * 

This is Newton's definition of curvature. 

Just as a straight line touching a curve, may be regarded as 
a line drawn through two points of the curve infinitely close to 
each other (definition of tangent), so a circle in contact with a 
curve may be considered to pass through three consecutive points 
of the curve infinitely near each other. Such a circle is called an 
osculatory circle or a circle of curvature. The osculatory circle 
of a curve has the same curvature as the curve itself at the point 
of contact. The curvature of different parts of a curve may be 
compared by drawing osculatory circles through these points. If 



< ;.;. FUNCTIONS WITH SINGULAR PROPERTIES. 141 

r be the radius of an oscillatory circle at P (Fig. 77) and r l that at 

P n then 

1 l 

curvature at P : curvature at P l = - : - . (3) 




In other words, the curvature at any two points on a curve varies 
ini-fTsely as the radius of the 
osculatory circles at these points. 
The radius of the osculatory 
circle at different points of a 
curve is called the radius of cur- 
vature at that point. The centre 
of the osculatory circle is the 
centre of curvature. F IG- 77. 

To find the radius of curvature of a curve. Let the coordinates 
of the centre of the circle be a and b, R the radius, then the 
equation of the circle is (page 76) 

(x - a)* + (y - bY = B 2 . x (4) 
Differentiating this equation and dividing by 2, f " 

dx 'u^'iu\> 

Again differentiating, 

i -L. / _ h\ d 'y , /^\ 2 _ n 



Let u = dy/dx and v = d' 2 yjdx 2 , for the sake of ease in manipulation, 
then (6) becomes 

y - b = - ^ . (7) 

Substituting this value of y - b in (5), 

1 + u 



a; - 



w, v, x and ?/ at any point of the curve are the same for the 
osculating circle at that point, and therefore a, b* and r can be 
determined from x, y, u, v. Substituting (7) and (8) in (4), 



R 



* The determination of a and b is of little use in practical work. They give 
equations to the evolute of the curve under consideration. The evolute is the curve 
drawn through the centres of the osculatory circles at every part of the curve, the 
curve itself is called the involute. Example : the osculatory circle has the equation 
( x - a ) z + (y - b) z = R. a and b may be determined from equations (4), (7) and (8). 

The evolute of the parabola y 2 = mx is 6 2 = (^ a ~ lfl P- 



142 HIGHER MATHEMATICS. 67. 

which is the standard equation. From (1), 

1 da d*yl(, (dy\*Y (-, /%\ 2 P7<% 

B - ds - a*/! 1 + U) } ; or * = I 1 + j/3' < 10 > 

EXAMPLES. (1) Find the radius of curvature at any point on the ellipse 
x*la* + 7/ 2 /6 2 = 1. 

dy = _ tto A , = _ * B = _ (ov 6V) I KM . 
- da? a 2 ?/' dx 2 a 2 y* 

At the point x = a, y = 0, R b 2 /a. Hint. The steps for d^y/dx 2 are : 
_ ft 2 y - x.dyjdx _ _ fe 2 o y + 6% 2 _ _ fe 2 a 2 ^ 2 
a 2 *' ?/ 2 a 2 * aV ~ a?'~if' 



(2) The radius of curvature of xy a, is (a: 2 + 



When the curve is but slightly inclined to the #-axis, dyldx 
is practically zero, and the radius of curvature is given by the 
expression 



a result frequently used in physical calculations involving capil- 
larity, superficial tension, theory of lenses, etc. 

The direction of curvature has been discussed in 59. It -was 
there shown that a curve is concave or convex at a point (x, y) 
according as d 2 y/dx 2 > or < 0. See also 100. 

67. Envelopes. 

m 
The equation y = + ax, 

represents a family of curves, since for each value of a we get 
a distinct curve. If a varies continuously it will determine a 
succession of curves, each of which is a member of the family 
denoted by the above equation, a is said to be the variable 
parameter of the family, since the different members of the 
family are obtained by assigning arbitrary values for a. Let the 
equations 

yi = + ax (i) 



(3) 



FUNCTIONS WITH SINGULAR PROPERTIES. 



143 



be three successive members of the family. As a general rule two 

distinct curves in the same family will have a point of intersection. 

Let P (Fig. 78) be the point of 

intersection of curves (1) and (2) ; 

P l the point of intersection of 

curves (2) and (3), then, since 

P l and P 2 are both situated on 

the curve (2), PP l is part of the 

locus of a curve whose arc PP 1 

coincides with an equal part of 

the curve (2). It can be proved, 

in fact, that the curve P P l . . . 

touches the whole family of 

curves represented by the original 

equation. Such a curve is said to 

be an envelope of the family. 

To find the equation to the FlG - 78. Envelope. 

envelope, bring all the terms of the original equation to one side, 

m 
y --- ax = 0. 

Then differentiate with respect to the variable parameter, and put 

m 

" x = ' 




Eliminate a between these equations, 



~ Jm.x - x = 0, or y - 2 



envelope 



Ti = 0. 



EXAMPLES. (1) Find the envelope of the family of circles 

(a- - a) 2 + y- = ?- 2 , 

where a is the variable parameter. Differentiate with respect to a and oc - a = ; 
eliminating a, we get y = + a, 
which is the required envelope. 
The envelope y = a represents 
two straight lines parallel to the 
x-axis and at a distance + a and 
- a from it. Shown Fig. 79. 

(2) Show that the envelope of 
the family of curves (x - m - a) 2 + 
y 2 = 4ma, is a parabola y 2 = 4??w. 

See 126 and 138. 



+0, 




envelop F 

FIG. 79. Double Envelope. 



144 



HIGHER MATHEMATICS. 



68. 



68. Six Problems in Maxima and Minima. 

It is first requisite, in solving problems in maxima and minima, 
to .express the relation between the variables in the form of an 

algebraic equation, and then to proceed 
as directed on page 130. 

In the majority of cases occurring 
in practice, it only requires a little 
common-sense reasoning on the nature 
of the problem, to determine whether a 
particular value of x corresponds to a 
maximum or to a minimum. 

(1) Divide a line into any two parts 
such that the rectangle having these two 
parts as adjoining sides may have the 
greatest possible area. 

If a be the length of the line, x the 
length of one part, a - x is the length of the other. The area of 
the rectangle will be 

y = (a - x)x. 




Differentiate and 



dv 

~r = a - 2x. 
dx 



Equate to zero, and x = ^a, that is to say, the line a must be 
divided into two equal parts, and the greatest possible rectangle 
is a square. 

(2) Find the greatest possible rectangle that can be inscribed in 
a given triangle. 

In Fig. 80, let b denote the length of the base of the triangle 
ABC, h its altitude, x the altitude of the inscribed rectangle. We 
must first find the relation between the area of the rectangle and 
of the triangle. By similar triangles, page 490, 

AH : AK = BC : DE ; h : h - x = b : DE, 
but the area is obviously y = DE x KH, and 

DE = jfa - x), KH = x ; .-. y = jj(hx - x 2 ). 

Now b/h is constant, and it is the rule, when seeking maxima and 
minima, to abbreviate the process by omitting constant factors, since, 
whatever makes the variable hx - x 2 a maximum will also make 

^(hx - x 2 ) a maximum. This is easily proved, for let 

y = C /M 



S r,s. FUNCTIONS WITH SINGULAR PROPERTI B& 145 

where c has any arbitrary constant value. For a maximum or 
minimum value 

dy/dx = cf(x) = 0, 
and this can onlv occur where 

f(x) = 0. 

Now differentiate the expression obtained above, for the area of the 
rectangle, and equate the result to zero. 

dy 

^ x = h - 2x = ; or x = \h. 

That is to say, the height of the rectangle must be half the altitude 
of the triangle. 

(3) To cut a sector from a circular sheet of metal so that the 
remainder can be formed into a conical- 
shaped vessel of maximum capacity. 

Let ACS (Fig. 81) be a circular plate of 
unit radius, it is required to cut out a portion 
AOB such that the conical vessel formed 
by joining OA and OB together may hold 
the greatest possible amount of fluid. We 
must again find a relation between the 
dimensions of the plate and the volume of 
the cone. 

Obviously ACB will be the circumference of the circular base 
of the cone. Let r denote the radius of this base, and y the 
perimeter of the circular base. 

y = 2irr ..... (1) 

If h denotes the height of the cone its volume F will be formula 
(26), page 492, 

F = \Trr*h ..... (2) 
But h and r form a right-angled triangle with hypotenuse 

OB = OA = 1, 
and whose base is r. 




h = v1 - r* ; from (1) h = l - y 2 l^'\ . (3) 
from (2) and (3) F = y* ^(1 - y*/4**)ll%ir. . . (4) 

The problem therefore is to find y such that F is a maximum. 
As before, omitting the constant term 1/127T, 



where F' x l/127r = F. Multiply through with Jl - 

a* 
K 



146 HIGHER MATHEMATICS. 68. 

divide through by y, since y is not zero, 

2 - 32/ 2 /47r 2 = ; or y = 2* ^2/3 . . (5) 
But, by Euclid vi., 33, 

perimeter of sector ACS : whole perimeter of the original circle 

= angle x : 360. 
Since the original circle had unit radius 

y : 27r = x : 360. 
Substituting this value of y in (5), 

x = 360 x/f = 294 (approx.). 

The angle of the removed sector is then about 66. The application 
to the folding of filter papers is obvious. 

(4) At what height should a light be placed above my writing table 
in order that a small portion of the surface of the table, at a given 

horizontal distance away from the foot 
of the perpendicular dropped from the 
light on to the table, may receive the 
greatest illumination possible ? 

Let 8 (Fig. 82) be the source of 
illumination whose distance from the 
j table x is to be determined in such a 
way that B may receive the greatest 
illumination. Let AB = a, and a the 
angle made by the incident rays SB = r on the surface B. 

It is known that the intensity of illumination varies inversely 
as the square of the distance of B, and directly as the sine of the 
angle of incidence. 

Since r 2 = a 2 + x 2 , siri a = x/r = x/' J(a 2 + x 2 ). 
In order that the illumination may be a maximum, 
.-. y = x/r 2 *J(a 2 + x 2 ) = x/ ^(a 1 + x 2 ) 
must be a maximum. Hence 

The interpretation is obvious.* 

(5) To arrange a number of voltaic cells to furnish a maximum 
current against a known external resistance. 

Let the electromotive force of each cell be E, and its internal 
resistance r. Let H be the external resistance, n the total number 
of cells. 

* Note : Negative and imaginary roots have no physical meaning in this problem. 
See page 394. 



FUNCTIONS WITH SINGULAR PROPERTIES. 



147 



Assume that x cells are arranged in series and nix in parallel. 
The electromotive force of the battery is xE. Its internal resist- 
ance x 2 r/n. The current C is given by 



C 




dC 
dx 



E - -x* 
n 



E 



N 



M 



Equate dO/dx to zero and simplify, 

.-. R = r&ln. 

That is to say, the^ battery must be so arranged that its internal 
resistance shall be as nearly as possible equal to the external 
resistance. 

(6) Snell's Law of Refraction of Light Index of Refraction. 

Let SP (Fig. 83) be a ray of light incident at P on the surface 
of separation of the media M and M' ; let PB be the refracted 
ray in the same plane as the incident 
ray. If PN is normal (perpendicular) 
to the surface of incidence, then 
SPN = i is the angle of incidence, 
N'PR = r the angle of refraction. 
Drop perpendiculars from S and R 
on to A and B, so that SA = a, 
EB = b. Now the light will travel 
from S to R in the shortest possible 
time, with a uniform velocity different 
in the different media M and M'. At 
the point P, the ray passes through 
the surface separating the two media, 
let AP = x, BP = p - x. Let the 
velocity of propagation of the ray of light in the two media be 
respectively v and v' per second. The ray therefore travels from 
S to P in SP/v seconds, and from P to B in RP/v' seconds, and 
the total time occupied in transit from S to R is 

t = SP/v + RP/v f .... (1) 
In the triangles SAP and EBP 
SP- 



. (2) 

* This formula is identical with the one given in any text-book on electricity. 
Note : C is a maximum when its reciprocal is a minimum. This and all the preceding 
results should be tested for maxima and minima by means of the second differential 
coefficients. 




FIG. 83. 



V >, 


/a 2 + x 2 


v' \/b 2 + 


(p - X Y 






x 




p - x 




N/< 


I 2 + X 2 ' 




+ (p- 


x)* 






sin i v 










sinr~ v'' 







148 HIGHER MATHEMATICS. 68. 

Substituting these values in (1) and differentiating in the usual 
way we get 

dx = 
But sin i = 

From (2) 

This result shows that the sifcns of the angles of incidence and 
refraction must be proportional to the velocity of the light in the 
different media, in order that the light may pass from one point 
to another in the shortest possible interval of time. 

The ratio, sin i/sin r, therefore, is constant for the same two 
media. This constant is usually denoted by the symbol /x, and 
called the index of refraction. It is obvious that for the same two 
media the index of refraction is constant for light of the same 
wave-length. 

EXAMPLES. (1) The velocity of motion of a wave in deep water is 

v = x /(A/o, + a/A) 

where A denotes the length of the wave, a is a constant. Required the length 
of the wave when the velocity is a minimum. (N. Z. Exam. Papers.) Ansr. 
K = a. 

(2) The contact difference of potential (E) between two metals is a function 
of the temperature (0) such that 

E = a + b0 + c0 2 . 

How high must the temperature of one of the metals be raised in order that 
the difference of potential may be a maximum or a minimum, a, b, c are 
constants. Ansr. = - 6/2c. 

(8) Show that the greatest rectangle that can be inscribed in the circle 
x z + y2 _ r 2 j s a S q uar e. Hint. Area 4xy, etc. 

(4) If v be the volume of water at 0C., v the volume at 0C., then, 
according to the " Hallstrom's" formula for temperatures between and 30, 

v = v (l - 0-000057,5770 + 0- 000007, 5601 2 - 0-000000,03509^). 
Show that the volume is least and the density greatest when 3-92. 

(5) Kopp's formula for the volume of water at any temperature between 
and 25 is 

v = v (I - 0-000061,0450 + 0'000007,71830 2 - 0-000000,037340 3 ). 
Show that the temperature of maximum density is 4-08. 

(6) An electric current flowing round a coil of radius r exerts a force F on 
a small magnet whose axis is at some point on a line drawn through the 
centre and perpendicular to the plane of the coil. If x is the distance of the 
magnet from the plane of the coil, 

Show that the force is a maximum when x = Ar. 



68. FUNCTIONS WITH SINGULAR PROPERTIES. 149 

(7) Draw an ellipse whose area for a given perimeter shall be a maximum. 
Although the perimeter of an ellipse can only be represented with perfect 

accuracy by an infinite series (page 188), yet for all practical purposes the 
perimeter may be taken to be ir(x + y) where x and y are the major and 
minor axes. The area of the ellipse is z = ir.ry. Since the perimeter is to 
be constant, a = w(x + y) or y = ajir - x. Substitute this value of y in the 
former expression and z = ax - itx*. Hence, x = a/2ir when z is a maximum. 
Substitute this value of x in y = / - x, and y = a/2ir, that is to say, 
x = y = a/2ir, or of all ellipses the circle has the greatest area. 

Boys' leaden water-pipes designed not to burst at freezing temperatures, 
are based on this principle. The cross section of the pipe is elliptical. If the 
contained water freezes, the resulting expansion makes the tube tend to become 
circular in cross section. The increased capacity allows the ice to have more 
room without putting a strain on the pipe. 

(8) If A, B be two sources of heat, find the position of a point O on the 
line AB = a, such that it is heated the least possible. Assume that the in- 
tensity of the heat rays is proportional to the square of the distance from the 
source of heat. Let AO = x, BO = a - x. The intensity of each source of 
heat at unit distance away is a and ft. The total intensity of the heat which 
reaches O is 



- * T (a - xr 
Find dl/dx and dPI/dx*. This equation is a minimum when 

Since AO : BO= /a: / show that when I is a minimum, its actual (numerical) 
value may be found from I(min.) = ( /a + /0) 3 /a 8 . If a = /3 then # = a, and 
the numerical value of I(min.) = 8o/a 2 . 

(9) Rapp's equation for the specific heat of water between and 100 is 

v = 1-039935 - 0-0070680 + 0-000212550 2 - 0-000001540 3 , 

where the mean specific heat between and 100 is unity. Hence show that 
there is a minimum between = 20 and 30, and a maximum about 70. 
Volten's equation for the same property is 

<r = 1 - 0-00146255120 + 0-00002379810 2 - 0-000000107160 3 . 
Hence show that there is a minimum between 40 and 50, and a maximum 
about 100. 

In the working of the above examples, it will be found simplest to use 
a, 6, c ... for the numerical coefficients, differentiate, etc., for the final re- 
sult, restore the numerical values of , 6, c . . . , and simplify. Probably the 
reader has already done this. 



150 



CHAPTEK IV. 
THE INTEGRAL CALCULUS. 

The experimental verification of a theory concerning any natural 
phenomenon generally rests on the result of an integration. 

69. Integration. 

IN the first chapter, methods were described for finding the mo- 
mentary rate of progress of any uniform or continuous change in 
terms of a limiting ratio, the so-called " differential coefficient " 
between two variable magnitudes. The fundamental relation 
between the variables must be accurately known before one can 
form a quantitative conception of the process taking place at any 
moment of time. When this relation or law is expressed in the 
form of a mathematical equation, the " methods of differentiation " 
enable us to determine the character of any continuous physical 
change at any instant of time. These methods have been 
described. 

Another problem is even more frequently presented to the 
investigator. Knowing the momentary character of any natural 
process, it is asked : " What is the fundamental relation between 
the variables?" "What law governs the whole course of the 
physical change ? " 

In order to fix this idea, let us study an example. The con- 
version of cane sugar into invert sugar in the presence of dilute 
acids, takes place in accordance with the reaction : 

C 12 # 22 n + S Z = 2C 6 # 12 6 
(cane sugar). (invert sugar). 

Let x denote the amount of invert sugar formed in the time t ; 
the amount of sugar remaining in the solution will then be 1 - x, 
provided the solution originally contained one gram of cane sugar. 
The amount of invert sugar formed in the time dt, will be dx. By 
Wilhelmy's law (page 46), the velocity of the chemical reaction 






THE INTEGRAL CALCULUS. 151 

at any moment will be proportional to the amount of cane sugar 
actually present in the solution. That is to say, 

dx 

3 -*(l-*). (l) 

where k is the " constant of proportion " (page 487). The meaning 
of k is obtained by putting x = 0. Thus, dx/dt = k, or, k denotes 
the rate of transformation of unit mass of sugar. 

From (1), page 5, 

v = dx/dt, . (2) 

where v denotes the velocity of the reaction. This relation is 
strictly true only when we make the interval of time so short 
that the velocity has not had time to vary during the process. 
But the velocity is not really constant during any finite interval 
of time, because the amount of cane sugar remaining to be acted 
upon by the dilute acid is continually decreasing. For the sake 
of simplicity, let k = ^, and assume that the action takes place 
in a series of successive stages, so that dx and dt have finite 
values, say Bx and St respectively. Then, 

(amount of cane sugar transformed) 8x ,\ 

(interval of time) &t ' 

Let St be one second of time. Let ^ of the cane sugar present 
be transformed into invert sugar in each interval of time, at the 
same uniform rate that it possessed at the beginning of the interval. 

At the commencement of the first interval, when the reaction 
has just started, the velocity will be at the rate of 0-100 grams of 
invert sugar per second. This rate will be maintained until the 
commencement of the second interval, when the velocity suddenly 
slackens down, because only - 900 grams of cane sugar are then 
present in the solution. 

During the second interval, the rate of formation of invert 
sugar will be jV of the 0-900 grams actually present at the be- 
ginning. Or, 0-090 grams of invert sugar are formed during the 
second interval. 

At the beginning of the third interval, the velocity of the re- 
action is again suddenly retarded, and this is repeated every 
second for say five seconds. 

Now let &c lf &c 2 , . . . denote the amounts of invert sugar 
formed in the solution during each second (8t). Assume, for the 
sake of simplicity, that one gram of cane sugar yields one gram 
of invert sugar. 



152 HIGHER MATHEMATICS. 69. 

(Cane sugar transformed.) 
During the 1st second, Sx l = 0-100 
2nd te a = 0-090 
3rd 5^ = 0-081 
4th Sx 4 = 0-073 
5th 5*5 = 0-066 

Total, 0-410 

This means that if the chemical reaction proceeds during 
each successive interval with a uniform velocity equal -to that 
which it possessed at the commencement of that interval, then, 
0'410 gram of invert sugar would be formed at the end of five 
seconds. As a matter of fact, 0*3935 gram is formed. 

But 0410 gram is evidently too great, because the retardation 
is a uniform, not a jerky process. We have resolved it into a 
series of elementary stages and pretended that the rate of forma- 
tion of invert sugar remained uniform during each elementary 
stage. We have ignored the retardation which takes place from 
moment to moment. . If we shorten the interval and determine 
the amounts of invert sugar formed during intervals of say half a 
second, we shall have ten instead of five separate stages to sum 

up, thus : 

(Cane sugar transformed.) 
During the 1st half second, 8x x = 0*0500 
2nd 8z 2 = 0-0475 

3rd Sx 3 = 0-0451 

4th 8z 4 = 0-0429 

5th Sx 5 = 0-0407 

6th Sx,. = 0-0387 

7th Sx 7 = 0-0367 

8th Sx s = 0-0349 

9th Sx 9 = 0-0332 

10th 5<r 10 = 0-0315 

Total, 0-401 

The quantity of invert sugar calculated on the supposition 
that the velocity is retarded every half second instead of every 
second, corresponds more closely with the actual change. The 
smaller we make the interval of time the more accurate the result. 
Finally, by making &t infinitely small, although we should have 
an infinite number of equations to add up, the actual summation 
would give a perfectly accurate result. To add up an infinite 
number of equations is, of course, an arithmetical impossibility, 



$ IM). THE INTEGRAL CALCULUS. 153 

but, by the " methods of integration " we can actually perform 
this operation. 

x (sum of all the terms v . dt, between t = and t} = 5,) 

= v . dt + v . dt + v . dt + . . . to infinity. 
This is more conveniently written, 



Je 
v .dt. 
o 
The signs "2" and "$" are abbreviations for "the sum of 

all the terms containing . . . " ; the subscripts and superscripts 
denote the limits between which the time has been reckoned. 
The second number of the last equation is called an integral. 
" \f(x) . dx " is read " the integral of f(x) . dx ". 

When the limits between which the integration (evidently 
another word for " summation") is to be performed, are stated, the 
integral is said to be definite ; when the limits are omitted, the 
integral is said to be indefinite. The superscript to the symbol 
4 'S" is called the upper or superior limit; the subscript, the 

J V 2 
p .dv denotes the sum 
i 
of an infinite number of terms p . dv, when v is taken between the 

limits v 2 and i\. 

To prevent any misunderstanding, I will now give a graphic 
representation of the 
above process. Take 
Ot and Ov as coordin- 
ate axes (Figs. 84 and 
85). Mark off, along 
the abscissa axis, in- 
tervals 1, 2, 3, . . . , 
corresponding to the 
intervals of time St. 
Let the ordinate axis 
represent the veloci- 
ties of the reaction 
during these different 
intervals of time. Let 
the curve vbdfh . . . represent the actual velocity of the trans- 
formation on the supposition that the rate of formation of invert 
sugar is a uniform and continuous process of retardation. This 




154 HIGHER MATHEMATICS. 69. 

is the real nature of the change. But we have pretended that 
the velocity remains constant during a short but finite interval of 
time say Bt = I second. The amount of cane sugar inverted dur- 
ing the first second is, therefore, represented by the area valO 
(Fig. 84) ; during the second interval by the area bc%l, and so on. 

At the end of the first interval the velocity at a is supposed 
to suddenly fall to b, whereas, in reality, the decrease should be 
represented by the gradual slope of the curve vb. 

The error resulting from the inexact nature of this " simplifying 
assumption" is graphically represented by the blackened area vab\ 
for succeeding intervals the error is similarly represented by bed, 
def, ... In Fig. 85, by halving the interval, we have consider- 




7'0 2'5 2>0 2-6 3-0 3-3 fO f-6 5-0 Seconds 

FIG. 85. 

ably reduced the magnitude of the error. This is shown by 
the diminished area of the blackened portions for the first and 
succeeding seconds of time. The smaller we make the interval, .the. 
less the error, until, at the limit, when the interval is made infinitely 
small, the result is absolutely correct. The amount of invert sugar 
formed during the first five seconds is then represented by the 
area vbdf ... 50. 

The above reasoning will repay careful study ; once mastered, 
the "methods of integration " are, in general, mere routine work. 

The operation denoted by the symbol " J "* is called integra- 
tion. When this sign is placed before a differential function, say 

*The symbol "J" is supposed to be the first letter of the word "sum". The 
first letter of the differential dx is the initial letter of the word "difference ". 



.* ;.. THE INTEGRAL CALCULUS. 100 

dx, it means that the function is to be integrated with respect to 
dx. Integration is essentially a method for obtaining the sum of 
an infinite number of infinitely small quantities. 

Not only can the amount of substance formed in a chemical 
reaction during any given interval of time be expressed in this 
manner, but all sorts of varying magnitudes can be subject to a 
similar operation. 

The distance passed over by a train travelling with a known 
velocity, can be represented in terms of a definite integral. The 
quantity of heat necessary to raise the temperature (6) of a given 
mass (m) of a substance from l to 2 , is given by the integral 

I 3 wcr . dO, where er denotes the specific heat of the substance. 

The ivork done by a variable force (F) when a body changes its 

ft 

position from s to s l is I F . ds. This is called a space integral. 

The impulse (magnitude of impressed force) due to a variable force 
F, acting during the interval of time 2 - t v is given by the time 

integral I F .dt. By Newton's second law, the change of mo- 
mentum of any mass (m), is proportional to the impressed force 
(impulse). Momentum is defined as the product of the mass into 
the velocity. If, when t is t lt v = v l and when t is t. 2 , v = v. 2 , 
Newton's law may be written 

m.dv = 



The quantity of heat developed in a conductor during the 
passage of an electric current of intensity i, for a short interval 
of time dt is given by the expression ki . dt (Joule's law), where k 
is a constant depending on the nature of the circuit. If the current 
remains constant during any short interval of time, the amount of 
heat generated by the current during the interval of time t. 2 - t lt 

Jt 2 
ki . dt. 
*i , 

The quantity of gas (q) consumed in a building during any 

interval of time t. 2 - t v may be represented as a definite integral, 



where v denotes the velocity of efflux of the gas from the burners. 
The value of q can be read off on the dial of the gas meter at any 



156 HIGHER MATHEMATICS. 69. 

time. The gas meter performs the integration automatically. 

The cyclometer of a bicycle can be made to integrate, s =\v . dt 

J ^i 
(v = velocity, t = time, s = distance traversed). 

Differentiation and integration are reciprocal operations in the 
same sense that multiplication is the inverse of division, addition, 
of subtraction. Thus, 

a x b -T- b = a; a + b - b = a. 
d\a .dx = a.dx', \dx = x. 

The differentiation of an integral, or the integration of a 
differential always gives the original function. The signs of 
differentiation and of integration mutually cancel each other. 
The integral, \f(x)dx, is sometimes called an anti-differential. 
Integration reverses the operation of differentiation and restores 
the differentiated function to its original value, but with certain 
limitations to be indicated later on. 

While any mathematical function can be differentiated without 
any particular difficulty, the reverse operation of integration is not 
always so easy, in some cases, it cannot be done at all. For in- 

f 2 f dx 

stance, the integrals 1 e x . dx and I -,. 8 , Y\ nave not vet 

J J Vv 2 * L ) 

evaluated. 

If, however, the function from which the differential has been 
derived, is known, the integration can always be performed. Know- 
ing that d(logx) = x~ l .dx, it follows at once that fa~ 1 .dx = logic. 

In many cases, we have to compare the integral with a tabu- 
lated list of the results of the differentiation of known functions. 
The reader will find it an advantage to keep such a list of known 
integrals at hand. A set of standard types is given in the next 
section, but this list should be extended. 

The Nature of Mathematical Eeasoning may now be defined 
with greater precision than was possible in 1. There, stress 
was laid upon the search for constant relations between observed 
facts. But the best results in science have been won by antici- 
pating Nature by means of the so-called working hypothesis. The 
investigator first endeavours to reproduce his ideas in the form of 
a mathematical equation representing the momentary state of the 
phenomenon.* Thus Wilhelmy's law (1850) is nothing more than 

* Mathematical equations containing differentials or differential coefficients, are 
called differential equations. 



70. THE INTEGRAL CALCULUS. i:>7 

the mathematician's way of stating an old, previously unverified, 
speculation of Berthollet (1779) ; while Guldberg and Waage's law 
(1864-69) is still another way of expressing the same thing. 

To test the consequences of Berthollet's hypothesis, it is clearly 
necessary to find the amount of chemical action taking place during 
intervals of time accessible to experimental measurement. It is 
obvious that Wilhelmy's equation in its present form will not do, 
but by " methods of integration " it is easy to show that if 



where x denotes the amount of substance transformed during the 
time t. x is measurable, t is measurable. We are now in a posi- 
tion to compare the fundamental assumption with observed facts. 

If Berthollet's guess is a good one, - . log - must have a con- 

L J- *~ X 

stant value. But this is work for the laboratory, not the study, 
as indicated in connection with Newton's law of cooling, 18. 

Integration, therefore, bridges the gap between theory and fact 
by reproducing the hypothesis in a form suitable for experimental 
verification, and, at the same time, furnishes a direct answer to the 
two questions raised at the beginning of this section. We shall 
return to the above physical process after we have gone through a 
drilling in the methods to be employed for the integration of ex- 
pressions in which the variables are so related that all the x's and 
dx's can be collected to one side of the equation, all the y's and 
dy's to the other. In Chapter VII., we shall have to study the in- 
tegration of equations representing more complex natural processes. 

If the mathematical expression of our ideas leads to equations 
which cannot be integrated, the working hypothesis will either 
have to be verified some other way,* or else relegated to the great 
repository of unverified speculations. 

70. Table of Standard Integrals. 

Every differentiation in the differential calculus, corresponds 
with an integration in the integral calculus. Sets of corresponding 
functions are called " Tables of Integrals ". 

*Say, by slipping in another "simplifying assumption". Clairaut expressed his 
ideas of the moon's motion in the form of a set of complicated differential equations, 
but left them in this incomplete stage with the invitation, " Now integrate them who 
can ". But see 107, 108, and 144. 



158 



HIGHER MATHEMATICS. 



The following are the more important ; handy for reference, 
better still for memorising : 



TABLE I. STANDARD INTEGRALS. 



Function. 


Differential Calculus. 


Integral Calculus. 




du n i 


/Wr ^ + 1 




da: nx 


J + 1" 




du 


//ja; 


u = a x . 


-7 a \os a. 
dx 


a^d = -^ . (2) 




du 


* 


u e x . 


-3 = e x - 
dx 


fe?dx = e*. . . (3) 


u = logx. 


du 1 


[dx \ 
J- = log x, or) 




du 1 1 


log a a- 


u log e a-. 


dx x e x 


log a e ' ' 




du 




u sin x. 




Jcos a-da- = sin x. . (5) 




du 




u cos x. 


-3- = - sin x. 
dx 


Jsin a?da- = - cos x. . (6) 




du 




u = tan x. 


-T- = secV. 


Jsec^da: = tan x. . (7) 




du 




u cot x. 


-5- = - -cosec^r. 


Jcosec 2 a- = - cot x. . (8) 


u sec x. 


du sin x 

CL3(s COS $ 


/srn^c^ seca;. . (9) 


u = .cosec x. 


du cos x 
dx ~ ~ sin'V 


/ cos x^ x _ _. coseca? _ (jo) 


u = sin ~ l x. 


dw 1 

dx ~ x /(l - a- 2 )' 
du 1 


, rfx | = sin-,. . (11) 
./x/^-^ 2 )! cos -^ (12) 




da- V(! - x ' 2 ) 




u = tan - l x. 


du _ 1_ \ 

da; ~ 1 + a- 2 ' 


^ ptan-^a-. . (13) 


u cot ^ J a-. 


du 1 


1 = - cot ~ l x. (14) 


do; ~ 1 + a- 2 ' ) 


w = sec ~ 1 a > . 


dw _ 1 
d^^ 1 


f dx i = 8^-^ (15) 
J a* N /(a- 2 - 1) ( cosec - ] .r (16) 




dx N /(a; 2 - 1) ) 






d^ 1 ) 


r vers ^ T (17) 


-M vers - l x. 


d# J(2x - x 2 )' ( 

d7^ 1 


r ^ 


u = covers - l x. 


da- N /(2a- - a- 2 ) 





71. The Simpler Methods of Integration. 

(1) Integration of the product of a constant term and a differ- 
ential. On page 24, it was pointed out that "the differential of 



$71. THE INTEGRAL CALCUU - 159 

the product of a variable and a constant, is equal to th constant 
multiplied by the differential of the variable". It follows directly 
that the integral of the product of a constant and a differential, is 
equal to the constant multiplied by the integral of the differential. 
X.g. t 

Ja . dx = a^dx = ax. 

Jlog a . dx = log a\dx = x . log a. 

On the other hand, the value of an integral is altered if a term 
containing one of the variables is placed outside the integral sign. 
For instance, the reader will see very shortly that while 

= ^x 3 ; x\xdx = ^x 3 . 



(2) A constant term must be added to every integral. It has 
been shown that a constant term always disappears from an 
expression during differentiation, thus, 

d(x + C) = dx. 

This is equivalent to stating that there is an infinite number 
of expressions, differing only in the value of the constant term, 
which, when differentiated, produce the same differential. In 
stating the result of any integration, therefore, we must provide 
for any possible constant term, by adding on an undetermined, 
"empirical," or "arbitrary" constant, called the constant of 
integration, and usually represented by the letter C. Thus, 

]du = u + C. 
If dy = dx, 

\dy + C l = \dx + C 2 ; 
y + C 1 = x + C 2 ; or, y = x + C, 
where C = C 2 - C r 

The geometrical signification of this constant is analogous to 
that of "6" in the tangent form of the equation of the straight 
line, formula (8), page 69 ; thus, the equation 

y = mx + b, 

represents an infinite number of straight lines, each one of which 
has a slope m to the z-axis and cuts the y-axis at some point b. 
An infinite number of values may be assigned to 6. Similarly, 
an infinite number of values may be assigned to C in $ . . . dx + C. 
According to Table I., 
dx dx 



etc. This means that sin ~ l x, cos ~ l x' r or tan - l x, cot ~~ l x, 



160 HIGHER MATHEMATICS. 71. 

only differ by a constant term. This agrees with the trigno- 
metrical properties of these functions illustrated in example (1), 
page 113. See also 106. The following remarks are worth 
thinking over : 

" Fourier's theorem is a most valuable tool of science, practical and theo- 
retical, but it necessitates adaptation to any particular case by the provision of 
exact data, the use, that is, of definite figures which mathematicians humorously 
call ' constants,' because they vary with every change of condition. A simple 
formula is n + n = 2n, so also n x n = n z . In the concrete, these come to the 
familiar statement that 2 and 2 equals 4. So in the abstract, 40 + 40 = 80, 
but in the concrete two 40 ft. ladders will in no way correspond to one 80 ft. 
ladder. They would require something else to join them end to end and to 
strengthen them. That something would correspond to a constant ' in the 
formula. But even then we could not climb 80 ft. into the air unless there 
was something to secure the joined ladder. We could not descend 80ft. into 
the earth unless there was an opening, nor could we cross an 80 ft. gap. For 
each of these uses we need something which is a ' constant ' for the special 
case. It is in this way that all mathematical demonstrations and assertions 
need to be examined. They mislead people by their very definiteness and 
apparent exactness. . . ." J. T. SPRAGUE. 

(3) Integration of a sum and of a difference. Since 

d(x + y + z + . . . ) = dx + dy + dz + . . . , 
it follows that 

$(dx + dy + dz + . . . ) = \dx + \dy + \dz + . . . , 

= x + y + z + . . . , 
plus the arbitrary constant of integration. 

It is customary to append the integration constant to the final 
result, not to the intermediate stages of the integration. 
Similarly, 

\(dx - dy - dz -...) = ]dx - \dy - \dz - . . . 
= x - y - z - . . . + C. 
EXAMPLES. (1) Show 
f{log (a + bx) (1 + 2x)}dx = Jlog (a + bx)dx + Jlog (1 + 2x)dx + C. 

(2) Show (logf^rl^* 4 J ! g ( a + bx ) (lx ~ Pg (* + 2 ^ dx + C - 

(4) Integration of x n . dx (see page 22). Since 

d(x n+l ) = (n + l)x n dx; x n .dx = dx n + l /(n + 1) ; 



C. . (1) 

To integrate any expression of the form ax n . dx, it is, therefore, 
necessary to increase the index of the variable by unity, multiply 



THE INTEGRAL CALCULI - 161 

bit <iny constant term that may be present, and divide the product 
by the new index. 

An apparent exception occurs when n = - 1, for then 

*~ 1 + 1 _ * _ 

See'page 224. We have seen, page 36, (6), that 

dx 
d(\ogx) = = x~ l .dx, 

.-. \x~ l .dx = logx + C. ... (2) 
If, therefore, the numerator of a fraction can be obtained by 
the differentiation of its denominator, the integral is the natural 
logarithm of the denominator. 

It is worth remembering that instead of writing log x + C, we 
may put 

log x + log c = log ex, 
for log c is an arbitrary constant as well as C. 

EXAMPLES. (1) Show ja . dx/bx = (a . logx)lb + C. 

(2) Show J2bx . dxj(a - bx*) = - log(a - bx*) + C. 

(3) Show jax? . dx = {ax* + C. 

(4) Show jax- ll5 .dx = 5ax*' 5 + C. 

(5) One of the commonest equations in physical chemistry is, 

dx = k(a - x) . dt. 

C dx 
Rearranging terms, kt = / - , 

.-. kt = - log (a - x), 
but log 1 = 0, 

.-. kt = log 1 - log(a - x), or, k = 7 lo g^r^ + c - 

(6) Wilhelmy's equation, 

dy fdy 

-V7 = - ay, may be written I = - at. 

Remembering that loge = 1, we have 

log y = log b - at log e ; or, log y = log e ~ at + log b, 
where log b is the integration constant, hence, 

log be ~ at = log y ; y = be - at . 
The meaning of these constants will be deduced in the next section. 

(7) By a similar method to that employed for evaluating Jx n dx, J* l dx, 
show 

\a x dx = j^ + C ; $e*dx = e* + C ; je~ "*dx =---. . (3) 
In the same way verify the results in Table I. 

(8) Prove - f^ = ^^ ^n + C, .... (4) 
by differentiating the right-hand side. Keep your result for use later on. 

L 



162 HIGHER MATHEMATICS. 72. 

(9) Evaluate Jsin 4 ; . cos x . dx. Note that coaxdx = d($inx), and that 
sin 4 x is the mathematician's way of writing (sin x)*.* 

.-. fsin 4 o; . cos x . dx (" sin 4 ic . d(sin x) = ^ sin 5 ie . + C. 

(10) What is wrong with this problem : " Evaluate the integral ja? 3 " ? 
Hint, the symbol " J " has no meaning apart from the accompanying " dx ". 
For brevity, we call " | " the symbol of integration, but the integral must be 
written, J . . . dx. 

(5) Integration of the product of a polynomial and its differ- 
ential. Bead (3), page 24. This is a simple extension of the 
preceding. Since 

d(ax m + b) H = n(ax m + b) n ~ l . amx m ~ l .dx, 

where amx m ~ l .dx has been obtained by differentiating the ex- 
pression within the brackets, 

.-. n\(ax m + b) n ~ l amx m ~ l .dx = (ax m + b) n + C. . (5) 

To integrate the product of a polynomial with its differential, 
increase the index of the polynomial by unity and divide the result 
by the new exponent. 

EXAMPLES. (1) Show J(3az 3 + l)*9ax z . dx = %(3ax s + I) 3 + C. 
(2) Show j(x + 1) -' . dx = B(x + I) 1 / 3 + C. 



(6) Integration of expressions of the 

(a + bx + ex 2 + . . . ) m xdx, . . (6) 

where m is a positive integer. Multiply out and integrate each 
term separately. 

EXAMPLES. (1) Show J(l + x)*x 3 dx = (| + x + %x*)x* + C. 
(2) J( + x^x^dx = (fa 2 + x% + *x)x% + C. 

The favourite methods for integration are by processes known 
as "the substitution of a new variable," "integration by parts" and 
by " resolution into partial fractions". The student is advised to 
pay particular attention to these operations. Before proceeding 
to the description of these methods, we shall return once more to 
the integration constant. 

72. How to find a Value for the Integration Constant. 

It is perhaps unnecessary to remind the. reader that integration 
constants must not be confused with the constants belonging to the 

* But we must not write sin ~ l x for (sin x) ~ l , nor (sin x) ~ l for sin ~ 1 x. Sin ~ l , 
cos ~ 1 , tan ~ l , . . . have the special meaning pointed out in 15. 



,< 7-J. THE INTEGRAL CALCULUS. 163 

original equation. For instance, in the law of descent of a falling 
body 

dc-dt = < i ; \dv = y\dt, or, v = gt + C. . . (1) 
Here (j is a constant representing the increase of velocity due 
to the earth's attraction, C is the constant of integration. The 
student will find some instructive remarks in 118. 

There are two methods in general use for the evaluation of the 
integration constant. 

FIRST METHOD. Eeturning to the falling body and to its 
equation of motion, 

v = gt + C. 

On attempting to apply this equation to an actual experiment, 
we should find that, at the moment we began to calculate the 
velocity, the body might be moving upwards or downwards, or 
starting from a position of rest. Ah 1 these possibilities are included 
in the integration constant C. Let v denote the initial velocity 
of -the body. The computation begins when t = 0, hence 
V Q = g x + C, or, C = v . 

If the body starts to fall from a position of rest, v = C = 0, 
and 

^dv = gt, or, v = gt. 

This suggests a method for evaluating the constant whenever 
the nature of the problem permits us to deduce the value of the 
function for particular values of the variable. 

If possible, therefore, substitute particular values of the vari- 
ables in the equation containing the integration constant and solve 
the resulting expression for C. 

EXAMPLE. Find the value of C in the equation 



which is a standard " velocity equation " of physical chemistry, t represents 
the time required for the formation of an amount of substance x. When the 
reaction is just beginning, x = and t = 0. Substitute these values of x and 
t in (2). 



Substitute this value of C in the given equation and we get 
t = !_/. 1 . 1\ 1 . a 



SECOND METHOD. Another way is to find the values of x 
corresponding to two different values of t. Substitute the two 



164 HIGHER MATHEMATICS. 73. 

sets of results in the given equation. The constant can then be 
made to disappear by subtraction. 

EXAMPLE. In the above equation, (2), assume that when t = t v x = x lt 
and when t = t 2 , x = x 2 ; where x lt a? 2 , ^ and t 2 are numerical measurements. 
Substitute these results in (2). 



By subtraction and rearrangement of terms 



The result of this method is to eliminate, not evaluate the constant. 

Numerous examples of both methods will occur in the course 
of this work. Some have already been given in the discussion on 
the " Compound Interest Law in Nature ". 



73. Integration by the Substitution of a New Variable. 

When a function can neither be integrated by reference to Table 
I., nor by the methods of 71, a suitable change of variable may 
cause the function to assume a less refractory form. The new 
variable is, of course, a known function of the old. 

This method of integration is, perhaps, best explained by the 
study of a few typical examples. 

(1) Evaluate J(a + x) n dx. Put a + x = y, therefore, dx = dy and 

J(a + x) n dx = ly n dy. 
From (1), page 158, 

\y n dy = y n ^l(n + 1) + C. 
Substitute for y, 

J(a + x) n dx = (a + x) n ^i(n + 1) + C. . . (1) 

EXAMPLES. Integrate the following expressions : 

(1) j(a - bx) n dx. Ansr. - (a - bx) n + l l(n + 1) + C. 

(2) |(a 2 + x 2 )-n*xdx. Ansr. ^(a 2 + z 2 ) + C. 

(3) J(a + x) ~ m dxl Ansr. - l/(m - 1) (a + x) m ~ 1 + C. Keep this result 
for future reference. 

Ansr. log (log a?) + C. 
logcc 

When the student has become familiar with integration he will find no 
particular difficulty in doing these examples mentally. 

(2) Integrate (1 - ax) m x n dx. Put y = 1 - ax, therefore, 
x = (1 - y)/a and dx = - dy/a. 



J7& TIIK IN TKGUAL CALCl'LUS. 165 

Substitute these values of x and dx in the original equation. 
5(1 - ax)'xdx = - ^nS(l - y)y m dy, 

which is the type of (6), page 162. The rest of the work is obvious. 
Method (6), 71. 

(3) Trignometrical functions can often be integrated by these 
methods. For example, required the value of $tan xdx. 

Jf sin x, 
tan xdx = \ dx. 
JGOSX 

Let cos x = u, - sin xdx = du. Since - \duju = - log u, and 
log 1 = 0. 

I tan xdx = log - - = log sec x + C. 
J 

EXAMPLES. Show that (1) Jsin x . cos x . dx = |sin 2 x + C. 

(2) J(l + xyx*dx = ^(1 + xY^(5x - x + |) + C. 

(3) Jcot xdx = log sin x + C. 

(4) Jsin x . dxjcos^x = sec x + C. 

(5) Jcos x . dxfsiv?x = - cosec x + C. 

(6) Evaluate je ~ * 2 xdx. Multiply and divide by - 2 

-\\e-^d(-x^=- &-**+ C. . . (2) 



dx 

(4) Integrate ,, 2 _ ^^ Put y = x/a, .\ x = ay, dx = ady, 

^(a* - ^ 2 ) = a s/1 -^ ; 

Jd# r % r 7 ^ 

-7=f =T 2 = -7= - 2 = ^in ^)= sin- !- + C. 
Va 2 - a? 2 J vl - i/ 2 \ a 

See page 166. 

EXAMPLE. Integrate . - ^dx, by substituting x = -.. 

Ansr. - N /(a 2 - 2 ) :{ . /3a 2 + C. 

(5) Some expressions require a little "humouring". Facility 
in this art can only be acquired by practice. A glance over the 
collection of formulae in Chapter XII. will often give a clue. In 
this way, we find that sin x = 2 sin \x . cos \x. Hence integrate 

fdx . f dx fsec x . dx 

srn^' *- e< j2sink;.cos^ or J 



Divide the numerator and denominator by cos' 2 i#, then, since 
l/cos 2 Ja? = sec' 2 ^ and d(tana?) = aec?x.dx, page 32, (3), 
~. f dx _ [sec 2 ^ . d($x) = rd(tan^) 
' J sin x ~ J tanja; ~J tan \x 
= log tan \x + C. 



166 HIGHER MATHEMATICS. 73. 

EXAMPLES. (1) Remembering that cos x = sin ($TT + x), (8), page 499, show 
that \dx]cos x = log tan (TT + %x) + C. 

(2) Integrate / sln j^ cosa .- Hint > see (17), page 499. 



f 
j 



cos 2 x + sin 2 # , /'cos x , /'sin # 
dx = 



snxcosx 

Here are a few useful though simple " tips " for special notice : 
1. Any constant term may be added to the numerator of a fraction pro- 
vided the differential sign is placed before it. The object of this is usually 
to show that the numerator of the given integral has been obtained by the 
differentiation of the denominator. If successful the integral reduces to the 
logarithm of the denominator. E.g., 



2. Jsin nx . dx may be made to depend 011 the known integral Jsin nx . d(nx) 
by multiplying and dividing by n. E.g., 

Jcos nx .dx - |cos nx . d(nx) sin nx + C. 

3. Add and subtract the same quantity. E.g., 

rx.dx c( X + a> - * (/i i i \ 

J l^Tx = J -1 + 2* dx = j U - 2 ' lT2i ) dx > etc ' 

4. Note the addition of log 1 makes no difference to the value of an ex- 
pression, because log 1 = 0; similarly, multiplication by log e e makes no 
difference to the value of any term, because log<,e = 1. 

(6) It very frequently happens that an expression involving the 
square root of a quadratic binomial can be readily solved by the 
aid of a lucky trignometrical substitution. The form of the in- 
verse trignometrical functions (Table I.) will sometimes guide us 
in the right choice. If the binomial has the forms : 

\/l - x 2 , or >/a 2 - x 2 , try x = sin 0, or a sin 0, or cos ; 

\/x 2 1, or Va? 2 - a 2 , try x = sec 0, or a sec 0, or cosec 6 ; 

>Jx* + 1, or \/x 2 + a 2 , try x = tan 0, or a tan 0, or cot 6. 
(Lamb's Infinitesimal Calculus, p. 184 ; Williamson's Integral 
Calculus, p. 73.) 

EXAMPLES. (1) Find the value of j\/( 2 - x*)dx. In accordance with the 
above rule, put x = a sin 0, .-. dx = a cos 6 . d6. 

- .-. J\/( 2 - x*)dx = a a /cos 2 0d0; 

and since 2 cos 2 = 1 + cos 20, (28), page 500, we may continue, 
= a 2 J(l + cos 26)de, 
= %a?(0 + i sin 20) ; 
but x a sin 0, = sin - l xja, and 

| sin 20 = sin . cos = sin X /(1 - sin-0), 



C. 



7:1 THE INTEGRAL CALCULUS. 107 






If the beginner has forgotten his "trig." he had better verify these steps 
from the collection of trignometrical formulae in Chapter XII. See also (8), 
70. 

sin ede c de i r de e / e^ 



cos$e _ 2cos 2 $0 _ I + cos 




(3) Show f ,. f" -> = log (a; + Vx 2 + 1) + C. Put x = tan 0. Note 

.' S \ X T *| 

that tan (r + $0) = tan + sec = x + v /(x 2 + 1) ; sec 6d8 = log (tan 6 + sec 0). 

(4) Show I ,.^_ 1 > = log (a: + N/x 2 "^!) + C. Put x = sec 0. 

(7) jf%e integration of x m ~ l (a + bx n ) p . dx. (See 76, below.) 

(i.) If p is a positive integer. Expand the binomial and treat 
as on page 162. 

(ii.) If p is fractional, say p = r/s ; 

(a) Let m/n be a positive integer. Substitute a new variable 
with an index equal to the denominator of the fractional index p, 
so as to make a + bx n = z*. Then proceed as follows : 

EXAMPLE. Evaluate J'x 5 (l + x*) l ?dx. Here m = 6 ; n = 2 ; p = J. Put 

1 + x 2 = * 2 , then, x 2 = s 2 - 1 ; 
.-. 5 = \/i + x 2 ; x. dx = z . dz. 
Substitute these values as required in the original expression, 



+ x 2 ) + 42(1 + x 2 ) + 35} + C. 

(b) Let m/n be a negative integer. 

EXAMPLE. Evaluate Jx - 4 (1 + x 2 ) " ll *dx. Here m = 3; n = 2 ; p = . 
Put 

1 + x 2 = 2 2 x 2 ; .-. x - 2 = z> - 1 ; .-. x - 4 = (z* - I) 2 ; .-. x = (z 9 - 1) - 1/2 ; 

Hence, jx - 4 (1 + x 2 ) ~ 1/2 dx = - j(z z - l)dz ; 

(c) m/n + p is integral. The last example comes under this 
head. 



168 HIGHER MATHEMATICS. $ 74. 

7$. Integration by Parts. 

On page 26, it was shown that 

d(uv) = vdu + udv. 
Now integrate both sides * 

uv = tydu + \udv. 

Hence, \udv =. uv - \vdu + C, . . (1) 

that is to say, the integral of udv can be obtained provided vdu 
can be integrated. This is called integration by parts. 

EXAMPLES. Evaluate the following expressions : 

(1) jxlogxdx. Put 

u = log x, I dv = x .dx; 
du = dx/x, I v = ^x 2 . 
Substitute in (1) 

fa .dv = \x log x .dx uv - fa . du, 

= $x 2 log x - \\x . dx = %x 2 log x - x 2 , 
= \x 2 (log a? - ) + C. 

(2) jx cos nx .dx Put 

u = x, I dv = {cos nx . d(nx)}jn ; 
du = dx, I v = (sin nx)jn. 
From (1), Ja? cos nx . dx = (x sin nx)/n - |(sin nx . dx)jn ; etc. 

(3) Show by "integration by parts " that 

jx 2 sin x . dx = (2 - x 2 ) cos x + 2x sin a? + C. 

In this example there are two integrations to be performed, first x 2 cos x . dx, 
and then x cos x . dx. 

(4) Solve the equation, 

dv = a(v - 2v) . dtf ^p, 

where , a, p are constant and v = when t = 0. Ansr. 
log - i log (t> - 2v) = a*/ ^/p. 

(5) \xe*dx = (x - 1)0* + C. Prove this. 

The selection of the proper values of u and v is to be determined by trial. 
A little practice will enable one to make the right selection instinctively. 
The rule is that the integral jv . du must be more easily integrated than the 
given expression. In this example if we take u = e x , dv = xdx, fa . du becomes 
$jx*e x dx, a more complex integral than the one to be reduced. The right 
choice is u = x, dv = e x dx. 

(6) Show jx 2 e*dx = (x 2 + 2x - 2)e x + C. 

(7) Evaluate by "integration by parts," {^(a 2 - x 2 )dz. Put 

u = ^(a 2 - a? 2 ), \dv = dx; 

du = x.dx I ^(a? - x 2 ), \ v = x. 




J7& THE [NTJ5GRAL CALCULUS. 169 

Transpose the last term to the left-hand side ; 

2jVa a - x 2 . dx = x >/a a - x a + a sin - l xla (page 158), 
a 2 sin - xa + * /x 2 - x 2 + C. 



75. Integration by Successive Reduction. 

A complex integral can often be reduced to one of the standard 
forms by the "method of integration by parts". By a repeated 
application of this method, complicated expressions may often be 
integrated, or else, if the expression cannot be integrated, the non- 
integrable part may be reduced to its simplest form. See examples 
<3) and (6), 74. 

EXAMPLES. (1) Evaluate Jic 2 cos nxdx. Put 

u = x 2 , I dv = {cos nx . d(nx}\ln ; 
du = 2xdx, I v = (sin nx)jn. 
Hence, from (1), 

/o; 2 sin nx 2 f 
arcos nxdx = --- / x sin nxdx. ... (1) 

Now put u = x, I dv = sin nx . dx ; 

du = dx\ v = - (cos nx)/n. 
Hence, |x sin nx . dx = - (x cos nx)jn - J( - cos nx . dx)/n, 

= (- x cos nx)jn + (sin nx)ln 2 . ... (2) 
Now substitute (2) in (1) and we get, 

# 2 sin . nx 2x cos nx 2 sin nx 



In this example, we have made the integral Jx 2 cosnx' . dx depend on that 
of x sin nx . dx, and this, in turn, on that of - cos nx . d(nx) t which is a 
known standard form. 

(2) Evaluate jor'cos x . dx. Put 

u = x 4 , I dv = cos xdx ; 
du = ix 3 dx, \ v = sin x. 
.'. Jorkjos xdx = a^sin x - ^a^sin xdx. 
In the same way, 

dja^sin xdx = 4x 3 cos x - 3 . 4 jo; 2 cos xdx. 
Similarly, 

3 . 4jx 2 cos xdx = 3 . 4 . x* sin x + 2 . 3 . 4jx sin xdx, 
And finally, 

2.3. 4jx sin xdx = 2 . 3 . 4x cos x + 1 . 2 . 3 . 4 sin x. 

All these values must be collected together, as in the first example. In 
this way, the integral is reduced, by successive steps, to one of simpler form. 
The integral Jar* cos xdx was made to depend on that of x 3 sin xdx, this, in 
turn, on that of x~ cos xdx, and so on until we finally got fcos xdx, a well-known 
standard form. 

It is an advantage to have two separate sheets of paper in working through 
these examples ; oh one work as in the preceding examples and on the other 
enter the results as in the next example. 



170 HIGHER MATHEMATICS. 76. 

(3) Integrate jx s e*dx. 



- 2jxe*dx), 
+ 2 . 3(xe* - je*dx), 
= (x s - 3x 2 + 6,r - 6)e* + C. 
(4) Integrate coa n xdx ; sin n xdx and sin m x . cos n xdx (see page 184). 



76. Reduction Formulae (for reference). 

In 74, we found it convenient to refer certain integrals to 
a " standard formula ". In 75, we reduced a complex integral 
to simpler terms by a repeated application of the same formula . 
Such a formula is called a reduction formula. 

The following standard reduction formulae are convenient for 
reference, others will be found in 79 and elsewhere. 

A. The integral x m (a + bx n )v.dx, may be made to depend on that of 
\x m - n (a + bx n )P + l . dx, through the reduction formula : 



x - + 1 (o + bx)f + l - a(m - n + l)fo?-(a + bx n )P. dx 
.dx = _^___J2! 

where m is a positive integer. This formula may be applied successively until 
the factor outside the brackets, under the integral sign, is less than n. Then 
proceed as on page 162. 

B. In A, m must be positive, otherwise the index will increase, instead of 
diminish, by a repeated application of the formula. Therefore, when m i& 
negative, transpose A and divide by a(m - n + 1). Thus, 

. (B) 



.- n 

where m is negative. 

C. Another useful formula diminishes the exponent of the bracketed term 
in the following manner : 

x m + l (a + bx)*> + anp\x m (a + bx*y - l dx 
\**(a + bxny . dx = - -^-XL-i- _ , . ( C> 

where p is positive. 

D. 1i p is negative , 

x m + l (a + bx")v + ] + (np + m + n + l)jx m (a + bx n }v + l dx 

Formulae A, B, C, D have been deduced by the method of integration by 
parts. Perhaps the reader can do this for himself. 

NOTE. Formula A decreases (algebraically) the exponent of the monomial 
factor while B increases the exponent of the same factor. Formula C decreases 
the exponent of the binomial factor while D increases the exponent of the 
binomial factor. 



;j 77 THE INTEGRAL CALCULI 9, 171 

EXAMPLES. Evaluate the following integrals : 

(1) J N '(a + j- 2 )rf;r. Hints, use C, Put m = 0, 6 = 1, n = 2, ;> = $. Ansr. 
tf&V(a + 2 ) + a log {x + v /(a + x 2 )}] + C. 

(2) j> 4 r/.i '/s (/'- - z 2 ). Hints, put ?n = 4, 6 = 1, n = 2, p = J. Use A 
twice. Ansr. {8a 4 sin x/a - x(2x 2 + 3 2 ) v '(a 2 - x 2 ) + C. 

(3) jx*dx I v '(l - x 2 ). Hint, use A. Ansr. - (x 2 + 2) N /(l - x 2 ) + C'. 

(4) J" v '( + 6x 2 ) - 3 dx. Ansr. x(a + bx) - '/> + C. Use D. 

(5) J a? /^_ a2 . i - e - J ^ ~ 3 ( - 2 + a- 2 ) - idx. Hint, use B. m = - 3, 6 = 1, 

Jx 2 - rt 2 1 X 

n = 2, = - Ansr. v -- + sec- 1-. 



77. Integration by Resolution into Partial Fractions. 

Fractions containing higher powers of x in the numerator than 
in the denominator, may be reduced to a whole number and a 
fractional part. Thus, by division, 

s? . dx / x \ , 

FTI = r - * + ^nr 8 - 

The integral part may be differentiated by the usual methods, 
but the fractional part must often be resolved into the sum of a 
number of fractions with simpler denominators, before integration 
can be performed. 

We know that A may be represented as the sum of two other 
fractions, namely 1 and ^, such that = 1 + ^. Each of these 
parts is called a partial fraction. If the numerator is a com- 
pound quantity and the denominator simple, the partial fractions 
may be deduced, at once, by assigning to each numerator its own 
denominator and reducing the result to its lowest terms. E.g., 
x 2 + x + 1 _2 ^ i 1 1 1 

~^ - ^3 + ^3 + ^3 ~ - x + 3,2 + ^3- 

When the denominator is a compound quantity, say -g , it 

x x 

is obvious, from the way in which the addition of fractions is per- 
formed, that the denominator is some multiple of the denominator 
of the partial fractions and contains no other factors. We there- 
fore expect the denominators of the partial fractions to be factors 
of the given denominator. Of course, this latter may have been 
reduced after the addition of the partial fractions, but, in practice, 
we proceed as if it had not been so treated. 

To reduce a fraction to its partial fractions, the first thing to 
do is to resolve the denominator into its factors and assume each 



172 HIGHER MATHEMATICS. 77. 

factor to be the denominator of a partial fraction. Then assign 
a certain indeterminate quantity to each numerator. These 
quantities may, or may not, be independent of x. The procedure 
will be evident from the following examples. There are four cases 
to be considered. 

Case i. The denominator can be resolved 'into real unequal 
factors of the type : 

1 

(a - x)(b - 'x)' ' 

Assume that 



(a - x) (b - x) a - x b - x' 

A(b - x) + B(a - x) 
(a - x) (b - x) ' 
1 _ Ab + Ba - Ax - Ex 

' ' (a - x)(b - x) ~ (a - x) (b - x) 

We now assume that the numerators on the two sides of this 
last equation are identical* and pick out the coefficients of like 
powers of x, so as to build up a series of equations from which 
A and B can be determined. For example, 

Ab + Ba = 1; x(A + B) = 0; .-. A + B = 0; .-. A = - B; 



- - , . . - - . 

b - a b - a 



* An identical equation is one in which the two sides of the equation are either 
identical, or can be made identical by reducing them to their simplest terms. E.g., 
ax 2 + bx + c = ax 2 + bx + c ; 

(a - x)l(a - x)* = \l(a - x), 
or, in general terms, 

a + bx + ex 2 + . . . =a' + b'x + c'x 2 + . . . 

An identical equation is satisfied by each or any value that may be assigned to the 
variable it contains. The coefficients of like powers of x, in the two members, are also 
equal to each other. Hence, if x = 0, a = a'. We can remove, therefore, a and a' 
from the general equation. After the removal of a and a', divide by x and put x = 0, 
hence b b' ; similarly, c c', etc. For fuller details, see any elementary textbook 
on algebra. 

The symbol " = " is frequently used in place of " = " when it is desired to em- 
phasise the fact that we are dealing with identities, not equations of condition. While 
an identical equation is satisfied by any value we may choose to assign to the variable 
it contains, an equation of condition is only satisfied by particular values of the vari- 
able. As long as this distinction is borne in mind, we may follow customary usage 
and write " = " when " = " is intended. For " = " we may read, " may be trans- 
formed into . . . whatever value the variable x may assume" ; while for " =," we 
must, read, "is equal to ... when the variable x satisfies some special condition or 
assumes some particular value ". See page 386. 






77 THE INTEGRAL CALCULUS. 173 

Substitute these values of A and B in (1). 

L_ _! L_ _J_ J_ (2} 

(a-x)(b - x) b - a a - x b - a b - x 
An ALTERNATIVE METHOD, much quicker than the above, is 
indicated in the following example : Find the partial fractions of 
the function in example (2) below. 

1 . * | B C . 

(a - x) (b - x) (c - x) a - x b - x c - x ' 

.-. (b - x) (c - x)A + (a - x) (c - x)B + (a - x) (b - x)C = 1. 

This identical equation is true for all values of x, it is, therefore, true 

when x = a, .', (b - a) (c - a)A = 1 ; .-. A = ^ _ a ^ c _ a ^ ; 
when*- 6, ,. (c - b) (a - b)B = 1 ; ,. B - (c . ^ , 6) ; 
when x = c, .-. (a - c) (6 - c)C = 1 ; .-. C = ^ _ c ^ b _ c j \ 



' (a - x) (b - x) (c - x) (b - a}(c - a) (a - x) 

1 1_ 

f (c - b) (a - b) (b - x) + (a - c) (b - c) (c - x)' 

EXAMPLES. (1) Show that 

t dx _ f dx r dx 

J (a - x) (b - x) ~ J (b - a) (a - x) ~ J (b - a) (b - x)' 

= , 1 . log _^^' + C. (3> 

b - a a - x 

(2) Evaluate / . Keep your answer for use later on. 

J (a - x) (b - x) (c - x) 

(3) Show that /".- *** 9 = JL log <L-^ + C. 

J a 2 - b z x 2 2ab 6 a - bx 

(4) J. J. Thomson's formula for the rate of production of ions by the 
Rontgen rays is 

!-)/(VM- 



Note that a - x 2 = (Ja - x)(Ja + x). 

(5) The velocity of the reaction between bromic and hydrobromic acida 
is, under certain conditions, represented by the equation : 

dx/dt = k(na + x) (a - x). 
Hence show that 

1 na + x 

~ (n + l)ar log a- x "* 

The constant is to be evaluated in the usual way by putting x = when 
t = 0. For practical convenience, this equation may be adapted for use with 
common logarithms by multiplying the right-hand side with 2-3026. 

dx = 2-3026 n(a+ x) 

(6) If ft = k(a + x) (na - 'x), show that k = . l]at . logio na _ x ,' 



174 HIGHER MATHEMATICS. 77. 

(7) Warder's equation for the velocity of the reaction between chloracetic 
acid and ethyl alcohol is 

dy/dx = ak{l - (1 + b)y] {!-(!- b)y}. 
Hence, show that 

lo glo [{l - (1 - %}/{! - (1 + %}] = 0-8686 abkt. 

Case ii. The denominator can be resolved into real factors 
some of ivhich are equal. Type : 

1 

(a - x) 2 (b - x) 
The preceding method cannot be used here because, if we put 

1 A B C _A + B C 

(a - x) 2 (b - x)~ a - x a - x b - x~ a - x b - x' 
A + B must be regarded as a single constant. Eeduce as before 
and pick out coefficients of like powers of x. We thus get three 
independent equations containing two unknowns. The values of 
A, B and C cannot, therefore, be determined by this method. To 
overcome the difficulty, assume that 

1 = A B C 

(a - x) 2 (b x)~ (a - x) 2 a - x b - x' 
Multiply out and proceed as before, thus, 

A l ' B l C l 

:= b-a ' b-a ' b-a' 

EXAMPLES. (1) Goldschmidt' 's equation for the velocity of the chemical 
reaction between hydrochloric acid and ethyl alcohol, is 

dxjdt = k(a - x) (b - ) 2 . 
Hence, 

, _ f dx _ I ( [ dx f dx f dx \ 

~ J (a -x)(b- x)* ~ a~^~b\J (b~=~xf ~l b^c ~J "="/' 



a - b b - x a - b b - x 
To find a value for C, put x = when t = 0. The final result is 

kt(a - 6j 2 = ^ ~ > x + log 



(2) Show 




(4) Price's equation for the velocity of the chemical reaction between 
hydrochloric acid and ethyl alcohol, is as follows: 

dxjdt = k{(a - x) (b - x)* - ax(c + x)(b - x)}. 

Integrate this equation and evaluate the constant for x = and t = 0. Ansr. 
Zablb + c)kt = * + c - frfl - 2) 1 x(a+b + ac 
VP *x(a +b + ac 

- Iog{x 2 (l - a) - x(a + b + acj + ab}/ab 



$77. THE INTKCiKAL CALCULI - 17:. 

where P = (a + b + oc) 2 - 4a6(l - a). This rather tedious example will } 
found in the Journal of tJie Clwnical ,SWiV///, 79, 314, 1901. 

(5) Walker tuul ./w/.st;/'.s equation for the velocity of the chemical reaction 
between hydrobromic and bromic acids, is 

dx\dt = k(a - x)*. 
Hence show that 3fc = {l/(a - .r) ;{ - 



The reader is probably aware of the fact that he can always 
prove whether his integration is correct or not, by differentiating 
his answer. If he gets the original integral the result is correct. 

Case iii. The denominator can be resolved into imaginary * 
factors all unequal. Type : 

1 

(a* + a?) (b + x)' 



* Imaginary Quantities. No number is known which will give a negative value 
when multiplied by itself. The square root of a negative quantity cannot, therefore, be 
a real number. In spite of this fact, the square roots of negative quantities frequently 
occur in mathematical investigations. Again, logarithms of negative numbers, inverse 
sines of quantities greater than unity, . . ., cannot have real values. 

Let V - 2 be such a quantity. If - a 2 is the product of 2 and - 1, + \/ - a 2 may- 
be supposed to consist of two parts, riz., + a and V - 1. Mathematicians have agreed 
to call a the real part of \/- a 2 and \/- 1, the imaginary part. Following Gauss, \/- 1 
is written i (or i). 

*J - 1, or i obeys all the rules of algebra. Thus, 
^Ti x v^l=-l; N /^4 = 2 N /rT; N /-^ x Jb= V56;i = V^T:i4 = l. 

EXAMPLES. (1) Show 

|4 = 1; i** + i = t; t - + 2 = _ i; 4 + 3 = _ tm < f ^ 

(2) Prove a 2 + 6 2 = (a + ib) (a - ib) . . . . (2) 

a + ib ac - bd be + ad 

(3) Show - = - 



(4) Show (a + ib) (c + id) = (ac - bd) + (ad + bc)t. 

(5) The quadratic x* + bx + c = 0, has imaginary roots only when b z - 4c is less 
than zero (formula (5), page 388). If a and j8 are the roots of this equation, show that 



a = - 6 + it V4C - 6 2 and ft = - \b - 
satisfy the equation. 

The imaginary numbers from - oo to +00 are : 

- OOt, . . ., -I, . . ., Ol, . . ., + i, . . ., + ODi, 

corresponding with the real numbers 

- oo, . . ., - 1, . . ., 0, . . ., + 1, . . ., 4- oo. 

By combining a real with an imaginary quantity we get what is known as a complex 
number, or a complex quantity. Such is x + ly. So important is the unthinkable 
J - 1 in modern theories, that the algebra of real quantity is now a special branch of 
the algebra of complex quantity. 

We know what the phrase "the point x, ?/" means. If one or both x and y are 
imaginary, the point is said to be imaginary. An imaginary point has no geometrical 
or physical meaning. If an equation is affected with one or more imaginary coefficients, 



176 HIGHER MATHEMATICS. 77. 

.' 

Since imaginary roots always occur in pairs (page 386), the 
product of each pair of imaginary factors will give a product of the 
form, x 2 + a?. Instead of assigning a separate partial fraction to- 
each imaginary factor, we assume, for each pair of imaginary 
factors, a partial fraction of the form : 

Ax + B 

aT+W 

Hence _ L Ax + B C 

= " 



EXAMPLES. Verify the following results 

dx C( A B Cx -:- D 



f dx C( A 

J (x-i)*(x* + i) ^J (w 



dx _ 1 1 1 + x 



Case iv. The denominator can be resolved into imaginary 
factors, some of which are equal to one another. Type : 



(a 2 + x*) 2 (b + x) 
Combining the preceding results, 

1 Ax + B . Cx + D E 



(a 2 + x*) 2 (b + x) (a* + z 2 ) 2 2 + & b + x 
In this expression, there are just sufficient equations to determine- 
the complete system of partial fractions, by equating the coefficients- 
of like powers of x. 

The differentiation of many of the resulting expressions usually 
requires the aid of one of the reduction formulae ( 76). 

EXAMPLE. Prove 

C(x A + x - l)dx _ f xdx f dx 
J (x 2 + I) 2 = )x* + 1 ~J(z 2 + I) 2 ' 
Integrate. Use formula D for evaluating the last term. 

Ansr. log (x 2 + 1) - Jar/(l + a- 2 ) + tan ~*x + C. 

the non-existent graph is conventionally styled an imaginary curve. Illustrations 
62 to 64. 

For a geometrical interpretation of >/ 1, see Lock's A Treatise on Higher Trig- 
nometry, 103, 1897 ; consult Chrystal's Algebra, Part I., Chapter XII., and Merriman 
and Woodward's Higher Mathematics, Chapter VI., for the algebra of complex numbers. 

Do not confuse irrational with imaginary quantities. In the former case, even if 
we cannot obtain the absolutely correct value, we can get as close an approximation 
as ever we please ; in the latter case, we cannot even say that the imaginary quantity 
is entitled to be called "a quantity ". 



THK IMK<;KAI. CALCULUS, 



177 



Cases iii. and iv. seldom occur in actual work. If, therefore, 
the denominator of any fractional differential can be resolved into 
factors, the differential can be integrated by one or other of these 
processes.. 

The remainder of this chapter will be mainly taken up with 
practical illustrations of integration processes. A few geometrical 
applications will first be given because the accompanying figures 
are so useful in helping one to form a mental picture of the opera- 
tion in hand. 



78. Areas enclosed by Curves. 
Integrals. 



To Evaluate Definite 



ft, 




1. To find the area bounded by two perpendiculars, dropped from 
any two points on a curve on to the x- (or y-) axis, the portion of 
the curve included betiueen these two points and the x- (or y-) axis 
included between the two perpendiculars. 

Let AB (Fig. 86) be any curve whose equation is known. It 
is required to find the area of the 
portion bounded by the curve, the 
two coordinates PM, QN, and 3/JV. 
The area can be approximately de- 
termined by supposing the portion 
PQMN cut up into small strips 
(called surface elements) perpen- 
dicular to the #-axis; find the area 
of each separate strip on the as- 
sumption that the curve bounding 
one end of it is a straight line and 
add the areas of all these trapezoi- 
dal strips together. (Cf. " Approxi- 
mate Integration," page 263.) 

Let the surface PQMN be cut up into two strips by means of 
the line LR. Join PR, RQ. 

(Area PQMN) = (Area PRLM) + (Area RQNL). 
But the area which is the sum of these two trapeziums is greater 
than that of the figure required, namely PrqQNM. The shaded 
portion of the diagram represents the magnitude of the error. 
It is obvious that the narrower each strip is made, the greater 
will be the number of trapeziums to be included in the calculation 



FIG. 86. 




178 HIGHER MATHEMATICS. g 78. 

and the smaller will be the error. If we could add up the areas 
of an infinite number of such strips, the actual error would become 
vanishingly small. Although we are un- 
able to form any distinct conception of 
this process, we feel assured that such an 
operation would give a result absolutely 
JR ff|fl g|J|||||||||| correct. ' But enough has been said on 
this matter in 69. We want to know 
how to add up an infinite number of 
infinitely small strips. 

In order to have some concrete image 
before the mind, let us find the area of 
PQNM in Fig. 87. In any small strip PESM, let PM = y, 
ES = y 4- 8y, OH = x and OS = x + &x. Let SA represent the 
area of the small strip under consideration. 

If the short distance PE were straight, not curved, the area A 
1 would be, (10), page 491. 

A = fa&x(PM + ES) &x(y + ^&y). 
By making $x smaller and smaller, the ratio, 

.approaches, and, at the limit, becomes equal to 

U _dA _ 

Or, dA = y . dx. . . . (1) 

In the same way, it can be shown that the differential of the 
area included between the curve and the y-axis, is, 

dA = x . dy (2) 

Formula (1), or (2), represents the area of an infinitely small 
strip. The total area (A) can be determined by integrating either 
of these formulae. For the sake of simplicity, we shall confine 
our attention to the former. But, before we can proceed any 
further, we must know the equation to the curve. 

(i.) Let rectangular coordinates be used. In any special case, 
the equation is to be solved for y, and the value of y so found is 
to be substituted in equation (1). Then integrate the resulting 
equation to get a general expression for an indefinite portion of 
the curve. To obtain the area of any definite portion situate 
between the ordinates of the extremities, we must take the sum 
of all the strips determined by the lengths of the ordinates. 



.< 78, THK INTWIRAL CALCULI - 17'.. 

For instance, the area of any indefinite portion of the curve, is 
A = \y.dx+ C ..... (3) 

and the area of the portion whose ordinates have the abscissae a. 2 
and a A (Fig. 86) is 

" (4) 



Equation (3) is an indefinite integral, equation (4), a definite 
integral. The value of the definite integral is determined by the 
magnitude of the upper and lower limits (see page 153). In Fig. 
86, if a lt a 2 , a a represent the magnitudes of three abscissae, such 
that a., lies between a x and a 3 , 

A = fry.dx + C = \' l y.dx + [\j .dx + C. 

fa J2 J3 

When the limits are known, the value of the integral is found 
by subtracting the expression obtained by substituting the lower 
limit in place of x, from a similar expression obtained by substi- 
tuting the upper limit for x. Thus, to evaluate \2xdx between the 
limits a and 6, 

Jit \b 

2x . dx = \ x 2 + C ; 
; |o 

or, as it is sometimes written, 

Par . dx = \x* + cT = (b* + C) - (a 2 + C) = V - a 2 . 

Plenty of examples will be given presently (see page 184). 

The process of finding the area of any surface is called, in the 
regular textbooks, the " Quadrature of Surfaces," from the fact 
that the area is measured in terms of a square. 

EXAMPLES. (1) To find the area bounded by an ellipse, origin at the 
centre. Here 

.,>* + y /6 = 1 ; or, y - 5 %/(rt a _ ^ 

liefer to Fig. 21, page 78. The sum of all the elements perpendicular to the 
.'-axis, from OPj to OP 4 , is given by the equation 





for. when the curve cuts the i--axis, .r = a, and when it cuts the y-axis, x = 0. 
The positive sign in the above equation, represents ordinates above the .r-axis. 
The area of the ellipse is, therefore, 



o 

Substitute the above value of y in this expression and we get for the sum of 
this infinite number of strips, 



180 HIGHER MATHEMATICS. 78. 

which may be integrated by parts, as shown on page 168, thus 
.4 = 4 -f^c v /( ft2 - a 8 ) + i 2 sin- llC + cT. 

The term within the brackets is yet to be evaluated between the limits x = a 
and x = 0. 

A = * 4W(<*' - a 2 ) + 2 sin-+ C - i<M a - O 2 ) + sin-'? + C 



.. - . 

a 2 

remembering that sin 90 = 1, sin - J l = 90 and 2 sin - l l = 180 = ir. The 
area of the ellipse is, therefore, irab. 

If the major and minor axes are equal, a = b and the ellipse becomes a 
circle whose area is TTO?. It will be found that the constant always disappears 
in this way when evaluating a definite integral. 

A WORD OF ADVICE. The student must learn to draw his own diagrams. 
If you are going to find the area bounded by a portion of an ellipse or of an 
hyperbola, first plot your curve. Squared paper is cheap enough. Carefully 
note the limits of your integral. 

(2) Find the area bounded by the rectangular hyperbola, 

xy a ; or, y ajx, 
between the limits x = ^ and x = x z . 



= a \ c + C = a|(log.r, + C) - (log ^ + C)}, 

= a log x<ilx r 

If a?! = 1 and x z = x, A = a log e x. This simple relation appears to be the 
reason natural logarithms are sometimes called hyperbolic logarithms. 

After this the integration constant is not to be used at any stage of the 
process of integration between limits. It has been retained in the above 
discussion to further illustrate the rule (see 72) : The integration constant of 
a definite integral disappears during the process of integration. Tlie absence of 
the indefinite integration constant is tlie mark of a definite integral. 

(3) Show that the area bounded by the logarithmic curve, x = log a, is 
y - 1. Hint. Evaluate C by noting that when x 0, y = 1, A 0. 

(ii.) Let polar coordinates be used. The differential of the area 
is then 

dA = .Vr 2 . dO. . . . . (5) 

EXAMPLE. Find the area of the hyperbolic spiral between and r. See 
5), page 96. 

rQ = a ; dQ = - a . drjr 2 . 



dA 



/o 10 

- I $a . dr = - I %ar = \ar. 

J -r "\- r 



2. To find tlie area enclosed between two different curves. Let PABQ and 
PA'B'Q (Fig. 88) be two curves, it is required to find the area PABQB'A'. 



78. 



THE INTK<;i!AI. CALCULUS. 



181 



Let /, =/i(.r) be the equation of one curve, ?/., = / 2 (#), the equation of the 
other. Find separately the areas PABQMN and PA'B'QMN, by preceding 
methods. The required area is, therefore, 

(Area PABQB'A') = (Area PABQMN) - (Area PA'B'QMN) 

= j . dx - |?/ 2 . dx. 

To find the area of the portion ABB' A', let x l be the abscissa of AR and x. t 
the abscissa of BS, then, 



A = 



(V . dr = P 2 (7/, - 
J*i J*i 



(6) 



EXAMPLE. Show that if the curves 

// 2 = 4z and 7/ 2 = 2a? - a: 2 , 
meet at the origin and at a point x = 8, y = 8, 



= 2/ ( v /2.r - .r a - \ f 4x)dx, etc. 

J 





-27T 



FIG. 89. 



3. The area bounded by two brandies of the same curve. If the curve is 
circular, 

y = v'O' 2 - -r 3 ), 

A = J\/(r 2 - x*)dx - \(- \V 2 - x*)dx, etc. 

4. To find the area bounded by tlie sine curve and the x-axis for a wlwle 
period (2), or for any number of wliole periods. Required the area OA * + ir JB2ir 
(Fig. 89). Let 

?/ = sin .T 
be the equation to the curve. 

A = I sin x . dx = - I cog 

= - (cos 2*-) - cos = 0, .... (7) 
for - cos2ir = - cos 360 = - 1 and cosO = 1. 

It can be shown in a similar manner that the area bounded by the cosine 
curve is zero. The geometrical signification of this will appear from Fig. 89. 

The instrument (electrodynamometer) used for measuring the strength of 
alternating electric currents, indicates the average value during half a com- 
plete period, that is to say, during the time the current flows in one direction. 
This is geometrically represented by the area of a rectangle Oafor, equal to the 
area of the portion bounded by the sine curve OAir and the x-axis. 



182 



HIGHER MATHEMATICS. 



Let the ordinate Afar be denoted by r and the height of the rectangle 
Oabir = Oa y ly then the area of OAv is 

In 
- cos x = ( - cos 180 -f cos 0)r = 2r, 
o 
since cos 180 = - 1, - cos 180 = 1. Therefore, 

r = 7/jir, represents the maximum current, y l the average current. 
Area of rectangle Oabir = area OAv ; or y^v = 2r. 
y l = 2r/ir = 0-6366r represents the average current. 

The maximum current is thus obtained by multiplying the average current 
by \TT, or by 1-5708. 







79. Graphic Representation of Work. 

Let a given volume (x) of a gas be contained in a cylindrical 
vessel in which a tightly fitting piston can be made to slide 

(Fig. 90). Let the sectional area 
X of the piston be unity. 

Now let the volume of the gas 
change dx units when a slight 
F IG- 90- pressure X is applied to the free 

end of the piston. Then by definition of work (W), 

Work = Force x Displacement; 
or, dW=X.dx. 

If p denotes the pressure of the gas and v the volume, we have, 

dW = p.dv. 

Now let the gas pass from one condition where x = x 1 to an- 
other state where x x> 2 . Let the corresponding pressures to 
which the gas was subjected be respectively denoted by X l and X z . 

By plotting the successive values of X 
and x, as x passes from x to a? 2 , we 
get the curve ACB, shown in Fig. 91. 
The shaded part of the figure represents 
the total work done on the system dur- 
ing the change. 

If the gas returns to its original 
state through another series of succes- 
sive values of X and x we have the 
FIG. 91. Work Diagram. curve ADB ( Fig> g 2 ) The total WQrk 

done by the system will then be represented by the area ABDx. 2 x r 
If we agree to call the work done on the system jwsitive, and work 
done by the system negative, then (Fig. 92), 



B 




T1IK IM I-:<JK.\L CALCULUS, 







IT, - W, = (An;, M'BXfBj - (Area 

= (Area ACBD). 

The shaded part in Fig. 92, therefore, represents the work done on 
the system during the above cycle of changes. A series of opera- 
tions by which a substance, after leaving a 
certain state, finally returns to its original 
condition, is called a cycle, or a cyclic 
process. A cyclic process is represented 
graphically by a closed curve. 

The reader will notice that the work 
is done on the system while x is increasing 
and by the system when x is decreasing. 
Therefore, if the curve is described by a 
point moving round the area ACBD in FlG " 92 --Work Diagram. ( 
the direction of the hands of a clock, the total work done on the 
system is positive ; if done in the opposite direction, negative. 

If the diagram has several loops, 
as shown in Fig. 93, the total work 
is the sum of the areas of the several 
loops developed by the point travel- 
ling in the same direction as the 
hands of a clock, minus the sum 
of the areas developed when the 
point travels in a contrary direction. 
This graphic mode of representing 
work was first used by Clapeyron. 
The diagrams are called Clapeyron's 



Work Diagrams. The subject is 
resumed on page 208. 




FIG. 93. Work Diagrams (after 
Clapeyron). 



80. Integration between Limits * Definite Integrals. 

It is perhaps necessary to further amplify the remarks on 
page 179. If f(x) denotes the first differential coefficient of /(#), 



, or, 



* Note the different meanings assigned to the word 
and in the integral calculus. 



limit" in the differential 



184 HIGHER MATHEMATICS. $ 80. 

EXAMPLES. (1) Show 1% -*. te --*"-'-*. 

.' (t 

(2) Prove [*_ a: 8 . dx = ${(3)* - ( - I) 3 } 9-?,-. 
One of the limits , or b, may become infinite or zero. 

(3) { e-*.dx =\ - e-*] = -e--(-e- Q ) = l, 
Jo. L -Jo 

since e ~ " = and e ~ = 1. 

f 06 dx 

(4) Show that / i -^/p + 1} = log(l + x /2). 

By way of practice verify the following results : 

/""/a I 17 / 2 

(5) I sin x . dx = - \ cos a; = - (cos TT - cos 0) = 1. 



o o 

pr/2 /TT/4 



-3) . . . 3.1 / Jr ' 8 , (?t- 1) (- 3) . . . 3 . 
)TT74T2~J ^ n(n-2).!.4.2 



pr/2 /TT/4 /. \ ,-JT 

(6) sin 2 * .dx = %*\ sin 2 * . da- = \ U- - 1 ; sin 2 * . dx = TT. 
/o Jo \* / J 

Hint for the indefinite integral. Integrate by parts. Put u = sin x, 
dv = sin x . dx. From (1), 74, 

fsin 2 ,r . dx = sin x . cos x + Jcos 2 j? . dx ; 

= sin x . cos x + |(1 - sin 2 x)dx. 

Transpose the last term to the left-hand side, and divide by 2. 
.-. |"sin 2 ,r . dx = $(sin x . cos x + x) + C. 

rir/2 n - 1 T 71 "' 2 

(7) I sin x .dx = / sin" - 2 x . d,c. 

J n J 

For n write 71 - 2 and show that 

r*v n _ 3 ,-ir/2 

/ sin " ~ 2 x . dx = ^o I sin " ~ *x . dx. 

Combine the last two equations and repeat the reduction. Thus, 
p/2 (n 

J 8 "fe= L 
when 71 is even ; 

f 2 . (71-1) (71 -8) ... 2 f^' 2 . (7Z.-l)(7t-3) ... 2 

Jo Sm ^ V=: n(n-2)./..3 J Sm < r ^- ^^27:7:3-' (2) 
when 71 is odd. 

(1) and (2) are useful reduction formulae. 

There are some interesting properties of definite integrals 
worth noting. 

(i.) It is evident that 

r/'(x)dx = /(a) - f(b) = -Ff(x)dx, . . (3) 

or, when the upper and lower limits of an integral are inter- 
changed, only the sign of the definite integral changes. This 
means that if the change of the variable from b to a is reckoned 
positive, the change from a to b is negative. That is to say, if 
motion in one direction is reckoned positive, motion in the 
opposite direction is to be reckoned negative. To put equation (3) 



< so. TIIK ENTEGRAL CALCULUS. 185 

in svords, the interchange of the limits of a definite integral causes 
the integral to chantje its siyn. 

(ii.) If m is any interval between the limits a and b. 

[ a f(x)dx = [f'(x)dx + [ m f(x)dx, . . (4) 

J& Jm Jft 

or, 

(iii.) If x is any function of a new variable y, so that f(x)dx 
becomes another function of y, say <j>'(y)dy, then, when x l and x 
are substituted for a?, y becomes y l and y. 2 respectively. 



l 

If a - y be substituted for x in this expression, 



[ U 

Jo 



f(x)dx = - (a - y)dy = ( - y)dy. 

J Jo 

But neither a? nor ?/ appears in the final result, hence we may put 
[f(x)dx = f/( - a?)da?. 

Jo Jo 

For instance, if/(#) = sin"a? ; /(^TT - a:) = cos";r, 

jr/2 pJT/2 

= I cos"xdx. ... (5) 



EXAMPLES. Verify the following results : 
(1) I '"ooeaxte = 1 ; 
From (1) and (2), if ?i is even, 



= f / 'n*r=( - !> (" - a > ' ' 3 v 1 .-. (6) 

Jo n(n - 2) ... 4 . 2 2 ' 



o 
and, if n is odd, 

- 8)... 4. 



o o n(n - 2) ... 5 . 3 

Test this by actual integration and by substituting n = 1, 2, 3, ... 

(2) f /!l 8inw?.r = l; /^W-d . d6 = 2 . 
J (i 32 J n 3 

If n is greater than unity, 

/ sin'".r . cos".rrf.r = ? \ "sin"'.r . cos" - 2 .rrf.r ; (8) 

J i, m + ?i./ 

if m is greater than unity, 

/ 'tin-He . co8 w a-da- = m ~ X / ^sin"- - V . cos"a-d.r. . . (9) 
Jo m + 11 J 

These important reduction formulae are employed in the reduction of 
either |'cos".rd.r, or fsin".rdr to an index unity, or zero. 

(3) I sin a- . cos xdx = ; / sin' J .r . cos xdx = J. 
.' o .' o 



186 HIGHER MATHEMATICS. si. 

/'T/2 -7T/2 

(4) I sin x . co$?xdx = A : ; / sin 2 x' . cos 2 a,YZx' = T Vir. 
./ Jo 

In the last integration, note cos a a; = 1 - sin 2 *. 

(5) Evaluate / sinmx. sin nxdx. By (26), page 499, 

J o 

2 sin nix . sin ?wj = cos(m - n).c - cos(w + ??)#. 
.. Jsin-ww; . sin nxdx = ^co$(m - n)xdx - JJcos(w + 7? 
_ sin(m - 7i)j; sin(w + n)x 
2(w -7ij~~ 2(r+~n)~" 
Therefore, if 7/1 and ?i are integral, 

,'ir 

I sin mx . cos nxdx . 0. 
Jo 
Remembering that sin TT = sin 180 = and sin = 0, if m = n, 

l f(l - cos 2nx)dx 
l 



(6) Show that / cos mx . cos nxdx is zero when m and 71 are integral ; 

J o 
J, when m = . Hints, cos TT = cos 180 - 1, cos - 1, 

2cos7;t,r . COS7M 1 = cosOw - n)x + coslm + n)x. 
(25), page 499. 

(7) Evaluate | asinfcc.cos Jz.dz. Ausr. 

J o 



2ft sn x . $n x = ?.a 

(8) / cos mx . cos nxdx = ; / sin nix . sin nxdx = ; 
J - it J * 



I cos mx . sin nxdx 0. 
/ it 

Hint. Use the results of Examples (5) and (6) ; also note that 
sin nxdx = - (cos nx)ln. 

(9) Show I co&O.dx = 2(a 2 - 6 2 sin 2 0)"cos e. 

For a more extensive treatment of definite integrals, the reader 
will have to consult some such work as that of Williamson, referred 
to elsewhere. 

81. To find the Length of any Curve. 

To find the length of any curve whose equation is known. 
This is equivalent to finding the length of a straight line of the 
same length as the curve, hence the process is called the " Recti- 
fication of Curves ". 

(i.) Let rectangular coordinates be used. It is required to find 
the length I of an arc AB (Fig. 94), where the coordinates of A 
are (o? , y Q ) and of B, (x tn y,,). Take any two points, P, Q, on the 



55 SI. 



TMK IM K<ii; Al. 



is: 



curve. Make the construction shown in the figure. Then by 
Euclid i., 47, if P and Q are sufficiently close, 
(Chord PQY = (to) 2 + (%)-'. 

But from (1), page 12, the limit of the chord PQ is equal to that 
of the arc PQ, 

.-. dl 



(dy)* ; or, = 



. dx. 



(1) 



The differential of an arc of any plane curve, referred to rect- 
angular coordinates, is equal to the square root of the sum of the 
squares of the differentials of the coor- 
dinates. 

In order to find the length of a curve, 
it is only necessary, therefore, to differ- 
entiate its equation and substitute the 
values of dx and dy, so obtained, in equa- 
tion (1). By integrating this equation, 
we obtain a general expression for the "o 
length of any arc. In order to find the 
length of any definite portion of the curve, we must integrate 
between the limits X Q and x, a or y and y n as the case might be. 

(ii.) Let polar coordinates be used. If the equation is 

f(0, r) = 0. 
The differential of the arc is 

dl = x /dr a + i*(dfff. ... (2) 




The rest is the same as before. 

EXAMPLES. (1) If the curve is a common parabola, 

7/2 = 4.r, 

.:ydy = 2adx, or (dx) 9 = y*(dy)*la 
From (1), dl = x '(?/ + 4rt 2 )f?7//2. 

Now integrate as on page 166, 

I =|7/\V + 4a 2 /rt + a\og(y + \ f f+~4a*) 
To find C, put y = when I = 0, 

C = - a log 2rt. 

* I = & \"V + 4fl2/ + a log \l(,j + N y + 
(2) Sliow that the perimeter of the circle 



C. 



is -2irr. Let / be the length of the arc in the first quadrant, then 

dy = x . dxjy. 



See page 166. 



4 x 



2irr. 



188 . HIGHER MATHEMATICS. 82. 

(3) Find the length of the equiangular spiral, page 96, whose equation is 

r = e9, or, 6 = logr/loge. 
Differentiate .'. do = drjr, .: dl = \ f 2.dr. 

.-. 2 = N/2.r + C; 
when r = 0, J = 0, C = 0, 

Z = \/2 . r. 

(4) Show that the cardioid curve, r = a(\ - cos 0) has I = 4a sin + C. 

(5) Show that the length of the cycloid, 

x = r(0 - sin 0) ; y = r(l - cos 0), 
from = to e = lf is 4r(cos0 - cos^). 

(6) Show that the length of the hypocycloid curve, 

2/3 + 02/3 _ r 2/3 ? i s 6r . 

Plot the curve. 

82. Elliptic Integrals. 

The ratio c/a (Fig. 21, page 78) is the eccentricity of the ellipse, the 'V 
of 44, page 95. Therefore (Fig. 21), 

c = ae ; but, c 2 = a 1 - b 2 , .-. 6 2 /a 2 = 1 - e 2 . 
Substitute this in the equation of the ellipse (7), page 79. Hence, 



Therefore, the length (I) of the arc of the quadrant of the ellipse (Fig. 21) is 



This expression cannot be reduced by the usual methods of integration. Its 
value can only be determined in an approximate way by methods to be 
described later on. 

Equation (1) can be put in a simpler form by noting that x = a sin <f>, 
where <f> is the complement of the " eccentric " angle 9 (Fig. 33). Hence, 



. d<b. 
o 

Here is called the amplitude and is written am u ; e, or, as it is sometimes 
written, k, the modulus of the function is always less than unity. 

The integral of an irrational * polynomial of the second degree, of the type, 

J *Ja + bx + ex 2 . X . dx ; or, J.X" . dx / \/a + bx + ex*- 

(where X is a rational function of a-), can be made to depend 011 algebraic, 
logarithmic, or on trignometrical functions, which can be evaluated in the 
usual way. But if the irrational polynomial is of the third or the fourth 
degree, the integral 

J *Ja + bx + cx' z + dx* + ex* . X . dx ; or, etc., 

cannot be treated in so simple a manner. Such integrals are called elliptic 
integrals. If higher powers than a; 4 appear under the radical sign, the re- 

* The numbers \/2~, v/5, . . ., which cannot be obtained in the form of a whole 
number or a finite fraction, are said to be irrational or surd ii inithci-s. On the contrary, 
\/4, N/27, |, 17, ... are said to be rational number*. 



82, THK INTEGRAL CALCULUa !*<> 

suiting integrals are said to be iiltni-t'lliptic or hyper -elliptic iu^v/-/i/.s. That 
part of an elliptic integral which cannot be expressed in terms of algebraic, 
logarithmic, or trignometrical functions is always one of three classes : 
1. Elliptic integrals of tlie first class : 



(2) 

since .c = sin <j>. This integral is used chiefly in the study of periodic oscilla- 
tions of large amplitude. For example, the time of a complete oscillation (t) 
of a simple pendulum of length /, oscillating through an angle a (less than 
180) on each side of the vertical is : 

/"T/2 fj,h 



t s 

where g is the constant of gravitation. We shall integrate this kind of 
equation in Chapter V. 

2. Elliptic integrals of tlie second class : 



rx . i c* /i M r 2 

E(k, 0) = I ^ N 'i _ & 5i^T2J ; or, E(k, x) = J A/-f ~z~ ' rf>r ' ' ^ 

just encountered in the rectification of the arc of the ellipse. 
3. Elliptic integrals of tfic third class : 

i'9p d<b 

n(n, k, <t>) = I \ - 7- T0 . ; or, n(, k, x)= etc., . (4) 

Jo (1 + n sm <f>) \ 1 - k z sin-ty 

where n is any real number, called Legendre's parameter. If the limits of the 
first and second classes of integrals are 1 and 0, instead of x and in the first case 
and ir/2 and in the second case, the integrals are said to be complete. Com- 
plete elliptic integrals of the first and second classes are denoted by the letters 
F and E respectively. \^1 - A^sin'fy is written A<ft- Since <f> = am u, x, 
the sine of the amplitude , is written x = sn it ; \/l - 2 = en it is the 
cosine of the amplitude of u and \ f l - k' z x' z = dn u, is the delta of the am- 
plitude of u. E.g., the centrifugal force (F) of a pendulum bob of mass (m) 
oscillating like the pendulum just described, is, 

where en t \ f gjl is the cosine of the amplitude of t*Jgfl^in the above elliptic 
integral (Class 1). 

There is a system of formulae connecting the elliptic functions to each 
other ; some of these have a certain formal resemblance to the trignometrical 
functions. Thus, 



d am ujdit = dfldu = v /(l - fc 2 sin 2 <f>) = dn u, etc. 

Legendre has calculated short tables of the first and second class of 
elliptic integrals ; the third class can be connected with these by known 
formulae. But numerical tables suitable for practical purposes are incom- 
plete.* 



* I learn from Baker's Elliptic lnti'ijrola that more complete tables ;uv in 
of computation. 



190 HIGHER MATHEMATICS. 83. 

Mascart and Joubert have tables of the coefficient of mutual induction of 
electric currents, in their Electricity and Magnetism (2, 126, 1888), calculated 
from E and F above. Greenhill's The Applications of Elliptic Function* 
{Macmillan & Co., 1892) is one of the most useful textbooks on this subject. 

83. The Gamma Function. 

It is sometimes found convenient to express the solution of a physical 
problem in terms of a definite integral whose numerical value is known, more 
or less accurately, for certain values of the variable. For example, there is 
"Legendre's table of the elliptic integrals ; Kramp's table of the integral 

e - ' 8 . dt ; Soldner's table of / dxllog x ; Gilbert's table of Fresnel's in- 



I 

/> /> 

tegral / cos %irv^ dv, or / sin \irv*. dv ; Legendre's table of the integral 
J o 

/ 



o o 

e - x x n ~ 1 . dx, or the so-called Gamma function, etc. 
By definition, the Gamma Function, or the Second Eulerian integral, is 

~oc 

T(n) = I e ~ *x n - l .dx ...... (1) 

This integral has been tabulated for all values of n between 1 and 2 to 
three decimal places. By the aid of such a table, the approximate value of 
all definite integrals reducible to Gamma functions can be calculated as 
easily as the ordinary trignometrical, or the logarithmic functions. There 
are three cases : 

1. n lies between 1 and 2. (Use Table II., 84.) 

2. n is a positive integer. (Use formula (4), below.) 

3. n is greater than 2. (Use (4) so as to make the value of the given 
expression depend on one in which n lies between 1 and 2.) 

Integrate the above integral by parts, thus, 

'/OB r* 

I e - x x" ,dx = nl e~ x x n - 1 . dx - e~ x x". . , (2) 
Between the limits x and ,r = GO, the last term vanishes. 

/GO ~<X 

Hence, I e - x x n . dx = n \ e~ x x ~ l . dx ; . (3) 

.' o J o 

or, T(n + 1) = nr(n) ..... (4) 

If n is integral, it follows from (4), that 

T(n + 1) = 1 . 2 . 3 . . . n = n ! . . . . (5) 

This important relation is true for any function of n, though n\ has a real 
meaning only when n is integral. 

The following are a few examples of the conversion of definite integral* 
into Gamma functions. For a more extended discussion the special text- 
books must be consulted. 

1. r(l) - 1 ; r(2) = 1 ; r(0) = oo ; r( - n) - oo ; r() = ^. . . (6) 

2. If a is independent of .r, 



~ : . dx = r(m) 



(8) 



THK INTK<;i;.\L CALCULI - IMl 

The first member of (8) is sometimes called the First F.iiler-Um Inteyml, or tin- 

t-'intction. It is written B(m, n). The Beta function is here expressed 

in terms of the Gamma function. Substitute x = ay/b in the second member 

r(w)r(n) 

f (ay + b)' + " a'"6'T(w + 71)' 




(10) 

If we substitute log .r in place of logf-r- 1 ), the expression on the right 
becomes 



xe-"*.dx = a-l m + Vr(n + 1) ....... (11) 



(13) 



EXAMPLES. Evaluate the following integrals: 
s 5 .r . 

r(f ) 



(1) I sin fi .r . cos 5 .r . dx. From (13), we may write this integral 



3 . 2 . i . . 



Mo Sinl - r -^ From ^' r( 6) 

\V I . ^ . f . f . ^ . x^ ,, 
2 ' 5.4.3.2.1 - - i" K ir 

r/C\ K / Q 

f/.r. Use (7). 
(5) If 



(4)/V 



i 



_ ( j ,. 



J 1 + a- ~ sin ;//./ 
show that F(w) . r(l - in) = ir/sin?;?.;- ; 

r(l + m) . r(l - 7) = ;ir/sin TT./ . 
Put w + 7^ = 1 in the Beta function, etc. 
(6) From the preceding result show that 

r(J) = '*. 
84. Numerical Table of the Gamma Function. 

When n has any value not lying between 1 and 2, the Gamma function 
r(u) may be readily calculated by means of equation (4), as indicated in the 
preceding examples. Table II., page 507, shows the value of 



log I e~ *x-- 1 . dx + 10, or, log r(n) + 



10, 



192 



HIGHER MATHEMATICS. 



; 86, 



to three decimal places for all values of n between 1 and 2. It has been 
abridged from Legendre's tables to twelve decimal places as they appear in 
his Exercices de Calcul Integral, tome ii., 80, 1817. 

Since T(n) is positive and less than unity for all values of n between 1 
and 2, logr(?i) will be negative for such values of n. Hence, as in the ordi- 
nary logarithmic tables of the trignometrical functions, the tabular logarithm 
is obtained by the addition of 10 to the natural logarithm of r(n). This must 
be allowed for when arranging the final result. 

85. To find the Area of a Surface of Revolution. 

A surface of revolution was defined, on page 108, to be a sur- 
face generated by the rotation of a line about a fixed axis, called 
the axis of revohition. 

Let the curve APQ (Fig. 95) generate a surface of revolution 
as it rotates about the fixed axis Ox. It is required to find the 
area of this surface. The quadrature of 
surfaces of revolution is sometimes styled 
the " Complanation of Surfaces ". 

Take any point P(x, y) on the curve. 
Let x receive an increment Sx = MN and 
y a corresponding increment By = QR. 
Draw PR and QS each equal to PQ and 
parallel to ON. Let s denote the area of 
the surface of revolution of the curve AP 
about the #-axis and 8s the surface generated by the revolution of 
PQ about the same axis. Let the length of the curve AP I and 
of the increment PQ = S/. 

If PR revolves about 02V, it will generate a cylinder whose 
superficies is 2irPM . PR (see page 491). QS revolving about ON 
will generate a cylinder whose surface is 2-n-QN . QS. Therefore, 
(Surface generated by QS) ZirQN . QS 



Q 




(Surface generated by PR) 



9xPJCiP.fi' 

y + Sy 



Therefore, 



Lt, 



ZirQN.QS 



y 



= i. 



But the surface generated by the arc PQ is intermediate between 
that generated by QS and by PR. Therefore, 

r .(Surface generated by PQ) _ ^ . ^ &s 
(Surface generated by PR) 2?^ 

- -^ ; or, ds = ^y . dl. . (I) 



= 1; 



THE INTEGRAL CALCULUS. 



From (1), page 187, dl-= J(dx)* + 



(dy) 2 , 



193 



.-. as = Uiry *J(dxy + (dyy. . . (2) 

If the curve revolves about the 7/-axis, similar formulae in x and 

y may be deduced. 

The reader may be able to reason out another way of obtaining 

the above result. See Figs. 98 to 100, page 195. 

EXAMPLES. (1) Find the surface generated by the revolution of the slant 
side of a triangle. Hints, equation of the line OC (Fig. 96) is y = mx, 
dy = mdx, 

ds 2iry v'l + w 2 . dx, 

s = \1irm x/1 + w 2 . xdx = irwa; 2 >/l + w 2 ' + C. 
Reckon the area from the apex, where x = 0, 
therefore C = 0. If x = h = height of cone 
OB and the radius of the base =r= BC, then, 
TO = rjh and 

s = -nWP~+T 2 = 2irr x ( Slant Height). 
This is a well-known rule in mensuration. 

(2) Show that the paraboloid surface generated by the revolution of the 
parabola, # 2 = lax, is %ira*{(a + z) 3 ' 2 - a 3 ' 2 }. 

(3) Show that the surface generated by the revolution of a circle is 4irr 2 . 




FIG. 96. 



86. To find the Volume of a Solid of Revolution. 

This is equivalent to finding the volume of a cube of the same 
capacity as the given solid. Hence the process is named the 
" Cubature of Solids ". 

The notion of differentials will allow us to deduce a method for 
finding the volume of the solid figure swept out by a curve rotating 
about an axis of revolution. At the same 
time, we can obtain a deeper insight into 
the meaning of the process of integra- 
tion. In order to calculate the volume 
of a body we may suppose it to be re- 
solved into a great number of elementary 
parallel planes, each plane being part of 
a small cylinder. Fig. 97 will, perhaps, 
help one to form a mental picture of the F IG- 97 (after Cox). 
process. It is evident that the total volume of the solid is the sum 
of a number of elementary cylinders about the same axis. If Sx be 
the height of one cylinder, y the radius of its base, the area of the 
base is Try 2 . But the area of the base multiplied by the height of 

N 




194 HIGHER MATHEMATICS. 87. 

the cylinder is the volume of each elementary cylinder, that is to 
say, 7ry 2 &x. The less the height of each cylinder, the more nearly 
will a succession of them form a figure with a continuous surface. 
At the limit, when Sx = 0, the volume of the solid is 

V=irly*.dx, ... (1) 

where x and y are the coordinates of the generating curve and the 
a?- axis is the axis of revolution. 

Formula (1) could have been obtained by a similar process of 
reasoning to that used in the preceding section. The abbreviated 
process here given illustrates how the idea of differentials facilitates 
the investigation of a complicated process. 

EXAMPLES. (1) Find the volume of the cone generated by the revolution 
of the slant side of the triangle in Example (1) of the preceding section. 

y = mx. 

dV = -n-y 12 . dx = irm^x' 1 . dx. 
.\V = $*m*x* + C. 

If the volume be reckoned from the apex of the cone, x = 0, and, therefore, 
C = 0. Let x = h and in = r/h, as before, 

(Volume of the entire cone) = %irr 2 h. 

(2) Show that the volume generated by the revolving parabola, y 2 = 4<7,r, 
is ^Try 2 x, where x = height and y radius of the base. 

(3) Required the volume of the sphere generated by the revolution of a 
circle, with the equation : 

a- 2 + y 2 ?- 2 . ( Volume of sphere) = f irr^. 

87. Successive Integration. Multiple Integrals. 

Just as it is sometimes necessary, or convenient, to employ 
the second, third or the higher differential coefficients d 2 y/dx 2 , 
d' 3 y/dx s . . . , so it is often necessary to apply successive integra- 
tion to reverse these processes of differentiation. 

(a) Successive integration with respect to a single independent 
variable. Suppose that it is required to reduce, d' 2 y/dx 2 = 2, to its, 
original or primitive form. We can write 



dx 2 dx\ 

.-. dy/dx = 2\dx = 2x + C P 
Again, dy = (2a? +' CJdx ; or, y = S(2# + CJdx, 

.-. y = x* + C^x + C 2 . 
In order to show that d 2 y/dx 2 is to be integrated twice, we write 

d 2 y = 2dx 2 , y =-- \\2dx 2 , or ^dx . dx. 
and hence, y = x 2 +. C^x + C.>. 



1 UK INTK<JRAL CALCULUS. 



L96 



Notice that there are as many integration constants as there are 
symbols of integration. 

Ansr. 



EXAMPLES. (1) Find the value of y = |'| !'"" <l-'"'. 
i ^ + \V\ X * + C<c + C*. 

(2\ Integrate d-s/df* = g, where g is a constant due to the earth's gravita- 
tion, t the time and s the space traversed by a falling body. 

.-. s = jjgr . dt' 2 = %gt 2 + C^t + C 2 , 

To find the values of the constants C l and C 2 . Let the body start from a 
position of rest, then, s = 0, t = C a = 0, C 2 = 0. See page 163. 

(b) Successive integration with respect 
to two or more independent variables. In 
finding the area of a curve, y = f(x), the 
same result will be obtained whether we 
divide the area Oab (Figs. 98 to 100) into 
a number of strips parallel to the a?-axis, 
as in Fig. 98, or vertical strips, Fig. 99. 
In the first case, the reader will no doubt FIG. 98. Surface Elements, 
be able to satisfy himself that the area A, 




in the second, 



= I x.dy; 

Jo 

fV 

Jo 



\ 






FIG. 99. Surface Elements. 



J 
dy for y in the last equation, 
o 

jv p 

^4=1 dx \ dy, 

Jo Jo 
which is more conveniently written, 

A = [ a ['dx.dy. 

Jo Jo 

This integral is called a double, or sur- 
face integral. It means that if we 
divide the surface into an infinite number 
of small rectangles (Fig. 100) and take 
their sum, we shall obtain the required FIG. 100. Surface Elements, 
area of the surface. 

To evaluate the double integral, first integrate with respect to 
one variable, no matter which, and afterwards integrate with 
respect to the other. x If x be taken first, we find the sum of all 
the rectangles formed bf the strips parallel to the .r-axis, that is to 
say, we integrate between the' limits a and 0, regarding dy as a 




196 HIGHER MATHEMATICS. 87. 

constant pro tern. ; we then take the sum of all the strips per- 
pendicular to the ic-axis, between the limits b and 0. 

When there can be any doubt as to which differential the limits 
belong, the integration is performed in the following order : the 
right-hand element is taken with the first integration sign on the 
right, and so on with the next element. 

rs ,-5 
EXAMPLES. (1) Evaluate I I x . dx . dy. 

Ansr. / x . dx\ y 1 = 3 / x . dx = 3 I \y? = 7*. 

J 2 L J 2 J 2 12 

(2) Show / " / xy 2 . dx . dy = ia 2 6 3 . 
J o J o 

In a similar manner, if the volume of a body is to be investi- 
gated, we obtain triple, or volume integrals by supposing the 
solid to be split up into an infinite number of little parallelepipeds 
along the three dimensions, x, y, z. These infinitesimal figures 
are called volume elements. The capacity of each little element 
dx x dy x dz. The total volume, or the volume integral of the 
solid is 



1- 



\dx . dy . dz. 

The first integration along the #-axis gives the area of an\ 
/ infinitely thin strip ; the integration along the ^/-axis gives the 
area of an infinitely thin portion of the surface, and a third in- 
\ tegration along the -axis gives the sum of all these little portions 
\of the surface, in other words, the volume of the body. 

In the same way, quadruple and higher integrals may occur. 
These, however, are not very common. Multiple integration 
rarely extends beyond triple integrals. 

EXAMPLES. (1) Evaluate the following triple integrals : 

/4 r5 re fi rz rG r4 ,-5 r6 

y0tdas.dy.da', \ yz* .dy .dz .dx; \ yz^ .dz .dx .dy. 

i J i J i J i J i J i J i J i J i 

Ansrs. 2580, 1550, 1470 respectively. 
(2) Show 



/" 

J J 



JO 

(3) Find the area (A) of the circle x 2 + y z = r 2 , and the surface (5) of the 
sphere x 2 - + 7/ 2 + z 2 = r 2 , by double integration. Ansrs. 

rr rv<r 2 -* 2 ) rr rv(>- 2 -* 2 ) dxdy 

4 -*/./. 



(4) 



r r /-vd 3 -* 2 ) r v (1 s - ** - 2 ) 

Evaluate 8 / / dx . dy . dz. Ansr. 

J o J o Jo 



88. THE INTEGRAL CALCULUS. 197 

Note sin Jir = 1. Show that this integral represents the volume of a 
sphere whose equation is x 2 + y 2 + z* = r 2 . Hint. The "dy" integration is 
the most troublesome. For it, put r 2 - z 2 = c, say, and use C, 76. As a 
result, y v/r 2 - x a - y* + (r 2 - .r 2 ) sin - l {y / \/r a - x 2 }, has to be evaluated 
between the limits y = ^/(r s - x 2 ) and y = 0. The result is -J(i-a - .r 2 )*-. The 
rest is simple enough. 



88. The Velocity of Chemical Reactions. 

The time occupied by a chemical reaction is dependent, among 
other things, on the nature and concentration of the reacting sub- 
stances, the presence of impurities and other " catalytic " agents, 
and on the temperature. 

With some reactions these several factors can be so controlled, 
that measurements of the velocity of the reaction agree with theo- 
retical results. 

A great number of chemical reactions have hitherto defied all 
attempts to reduce them to order. For instance, the mutual action 
of HI and HBrO^ of H. 2 and 2 , of carbon and oxygen and the 
oxidation of phosphorus. The magnitude of the disturbing effects 
of secondary and catalytic actions obscures the mechanism of such 
reactions. In these cases more extended investigations are re- 
quired to make deai* what actually takes place in the reacting 
system. But see 135 in Part II. Advanced. 

Fuhrmann (Zeitschrift fur physikalische Chemie, 4, 89, 1889) 
classifies chemical reactions into " orders " according as one or 
more molecules are included in the reaction. 

I. Reactions of the first order. Let a be the concentration of 
the reacting molecules at the beginning of the action when the 
time t = 0. The concentration, after the lapse of an interval of 
time t, is, therefore, a - x, where x denotes the amount of sub- 
stance transformed during that time. Let dx denote the amount 
of substance formed in the time dt. The velocity of the reaction 
at any moment is proportional to the concentration of the reacting 
substance (Wilhelmy's law), hence we have 

d j= k(a-x); or, k = \>^% c ~r x - - W 

where k is a constant depending on the nature of the reacting 
system. Eeactions which proceed according to this equation are 
said to be reactions of the first order. 



198 HIGHER MATHEMATICS. $ 88. 

II. Reactions of the second order. Let a and b respectively 
denote the concentration of two different substances in such a 
reacting system as occurs, when acetic acid acts on alcohol, or 
bromine on fumaric acid, then, according to the law of mass 
action, the velocity of the reaction at any moment is propor- 
tional to the concentration of the reacting substances. In this 
case 



Eeactions which progress according to this equation are called 
reactions of the second order. For the integration, see page 173. 

If the two reacting molecules are the same, then a = b. From 
(2), therefore, we get log 1 x 1/0 = x cc. Such indeterminate 
fractions are discussed on page 245. " It is there shown that when 
a = b, this expression may be made to assume the form, 

k = ' 



a(a - x)' ' 

This expression is also obtained by the integration on the corre- 
sponding equation, 

dx/dt = k(a - x) 2 . . . ' . . (4) 

Equation (4) is that required for reactions similar to the 
polymerization of nitrogen dioxide, etc. 



In the hydrolysis of cane sugar, 

CiAAi + H i = 2C 6 H 12 6 , 

let a denote the amount of cane sugar, b the amount of water 
present at the beginning of the action. The reaction is, therefore, 
represented by the equation, 

dx/dt = k'(a - x) (b - x), 

where x denotes the amount of sugar which actually undergoes 
transformation. 

If the sugar has been dissolved in a large excess of water, the 
concentration of the water is practically constant during the whole 
process. But b is very large in comparison with x, therefore, b - x 
may be assumed constant 

k = k'(b - x), 

where k' and k are constant. Hence equation (1) should represent 
the course of this reaction. 

Wilhelmy's measurements of the rate of this reaction show that 
the above supposition corresponds closely with the truth. 



$ss. THE INTEGRAL CALCULU8. 199 

EXAMPLE. Proceed as on page 43 with the following pairs of values of 
x and t : 

t = 15, 30, 45, 60, 75, ... 

x = 0-046, 0-088, 0-130, 0-168, 0-206, . . . 

Substitute these numbers in (1) ; show that k is constant. Make the proper 
changes for use with common logs. Put a = 1. 

The hydrolysis of cane sugar is, therefore, a reaction of the first 
order provided a large excess of water is present. 

III. Reactions of the third order. In this case three molecules 
take part in the reaction. Let a, b, c, denote the concentration of 
the reacting molecules of each species at the beginning of the 
reaction, then, 

dx/dt = k(a - x)(b - x)(c - x). . . (5) 
Integrate this expression as on page 173, put x = when t = 
, in order to find the value of C. The final equation can then be 
written in the form, 



,. - 

t(a - b) (b - c) (c - a) 
where a, b, c, are all different. 

This equation has been studied under various guises by Har- 
court and Esson, J. J. Hood, Ostwald, etc. (See the set of 
examples at the end of this section.) 

If we make a = b = c, in equation (5) and integrate the resulting 
expression 

dx - k(a - x)* k - 1 f 
~ ~ 



dt ~ ~ BlfT^ " ^J 2ta(a - xf 

The polymerization of cyanic acid is an example of such a 
change, 

3CNOH = C Z N 3 0. 3 H S . 
Rearrange the terms of equation (7) so that, 



x = a(l - II JZatkt + 1). ... (8) 
In order that we may have x = a, t must become infinite. This 
means that the reaction will only be completed after the elapse of 
an infinite time. 

If c = b in (6), and a is not equal to b, 

fc-i, * f (*-*)* + log *(*-*)! (9) 

t (a- b)*\b(b - x) ^b(a- x)j 

See examples at the end of this section. 

IV. Reactions of the fourth order. These are comparatively 
rare. The reaction between hydrobromic and bromic acids is, 



200 HIGHER MATHEMATICS. 88. 

under certain conditions, of the fourth order. So is the reaction 
between chromic and phosphorous acids (see page 175). 

The general equation for a reaction in which n molecules of the 
,same kind take part, is 

dx - k(a xY- 1 l 

~~ 



The intermediate steps of the integration are 



_ -L+ , n . n __ _ 



(n - I) (a - x) n ~ l (n - l)a n ~ 1 ' 

for, when x = 0, t = 0. 

To find the order of a chemical reaction. Let C l} C 2 be the 
concentration of the solution, that is to say, the quantity of re- 
acting substance present in the solution, at the end of certain 
times t and t 2 . From equation (10), 

" =kCn ' '''~' ^ 1 = kt + constant > 



where n denotes the number of molecules taking part in the re- 
action. It is required to find a value for n. From (11) 

= ^; or,n-l + logi/log'. . (12) 

t 2 j Gj 

Judson and Walker (Journal of the Chemical Society, 73, 410, 
1898) found that while the time required for the decomposition of 
a mixture of bromic and hydrobromic acids of concentration 77, 
was 15 minutes; the time required for the transformation of a 
similar mixture of substances in a solution of concentration 51 '33, 
was 50 minutes. Substituting these values in (12), 

* = 1 + 10 S 3 ' 333 = 3-97.* 
log 1-5 

The nearest integer, 4, represents the order of the reaction. 

The intervals of time required for the transformation of equal 
fractional parts m of a substance contained in two solutions of 
different concentration C l and C 2 , may be obtained by graphic in- 
terpolation (pages 68 and 254) from the curves whose abscissae 
are t l and t% and whose ordinates are C l and C 2 respectively. 

Another convenient formula for the order of a reaction, is 





The reader will probably be able to deduce this formula for himself 

* Use the table of natural logarithms, page 520. 



88. THE INTEGRAL CALCULUS. 201 

(see Noyes, Zeit.f. phys. Chem., 16, 546, 1895; Noyes and Scott, 
ibid., 18, 118, 1895). 

The mathematical treatment of velocity equations here outlined 
is in no way difficult, although, perhaps, some practice is still re- 
quisite in the manipulation of laboratory results. The following 
selection of typical examples illustrates what may be expected in 
practical work. The memoirs referred to may be considered as 
models of this kind of research. 

EXAMPLES. (1) It was once thought that the decomposition of phosphine 
by heat was in accordance with the equation, 4PJ3" 3 = P 4 + 6_H" 2 ; now, it is be- 
lieved that the reaction is more simple, viz., PH- = P + SH, and that the 
subsequent formation of the P 4 and H 2 molecules has no perceptible influence 
on the rate of the decomposition. Show that these suppositions respectively 
lead to the following equations : 



dx - ktl xV- - fc - - 1 1 

K( - X) ,..#=-. - - _ - 1. 

at t (1 x) 6 



Or, 



In other words, if the reaction be of the fourth order, k will be constant, and 
if of the first order k' will be constant. 

To put these equations into a form suitable for experimental verification, 
let a gram molecules of PH 5 per unit volume be taken. Let the fraction x 
of a be decomposed in the time t. Hence, (1 - x}a gram molecules of phos- 
phine and 3ax/2, of hydrogen remain. Since the pressure of the gas is pro- 
portional to its density, if the original pressure of PH 5 be p Q and of the mixture 
of hydrogen and phosphine p lt then, 

PilPo = {(l-x)a + 3xa/2}la = 1 + Jar, 
- 2 ; (1 - x)a = (3 - 




. 
t %t-fe 

Kooij (Zeit. f. phys. Chem., 12, 155, 1892) has published the following 
data : 

t = 0, 4, 14, 24, 46-3, . . . 

p = 758-01, 769-34, 795-57, 819-16, 865-22, . . . 
Hence show that k', not k, satisfies the required condition. The decomposi- 
tion of phosphine is, therefore, a reaction of the first order. 

(2) Does the reaction, 2PH 3 = 2P + 3// 2 , agree with Kooij 's observations? 
In experimental work in the laboratory, the investigator proceeds by the 

method of trial and failure in the hope that among many wrong guesses, he 
will at last hit upon one that will " go ". So in mathematical work, there is 
no royal road. We proceed by instinct, not by rule. E.g., we have here 
made three guesses. The first appeared the most probable, but on trial proved 
unmistakably wrong. The second, least probable guess, proved to be the one 
we were searching for. 

(3) Show that the reaction, 

CH^Cl . COOff + H 2 = CH, . COOff + SCI, 



202 HIGHER MATHEMATICS. 88. 

in the presence of a large excess of water is of the first order. See Van't Hoffs 
Studies in Chemical Dynamics (Ewan's translation), 130, 1896, for experi- 
mental work. 

(4) Find the order of the reaction between ferric and stannous chlorides 
from the two following series of observations : 

^ = 0, -75, 1, 1-5; I t 2 = 0, 1, 3, 7; 

x l x 10 = 10-0, 3-59, 4-19, 5'10 ; | x z x 10 = 6-25, 1'43, 2-59, 3-61, 
where x lt x% denote the amounts of ferric chloride reduced in the times t l and 
2 respectively. Use formula (13), put in and also m = ^. Ansr. Third. 

In the following examples, always verify your deduction by finding the 
numerical value of k when experimental data are given. 

(5) Reicher (Zeit. f. phys. Chem., 16, 203, 1895) in studying the action of 
bromine on f umaric acid, found that when t = 0, his solution contained 8-8 of 
fumaric acid, and when t = 95, 7*87 ; the concentration of the acid was then 
altered by dilution with water, it was then found that when t 0, the concentra- 
tion was 3-88, and when = 132, 3-51. Here dC l jdt = (8-88 -7'87)/95 = 0-0106 ; 
dC^dt = 0-00227 (page 200); C l =(8-88 + 7 -87)/2 = 8-375 ; C 2 =3'7, n=l-87 in 
(13) above. The reaction is, therefore, of the second order. 

(6) In the absence of disturbing side reactions, arrange velocity equations 
for the reaction, 

2C#" 3 . C0. 2 Ag + H . CO. 2 Na = CH 3 . COOH + CH 3 . CO^Na + CO 2 + 2Ag. 
Assuming that the silver, sodium and hydrogen salts are completely dissociated 
in solution, the reaction is essentially between the ions : 

Ag + H. COO = Ag + C0 2 + H, 

therefore, the reaction is of the third order. Verify this from the following 
data : When 

t = 2, 4, 7, 11, 17, . . . ; 

x x 10 3 = 62-25, 69-15, 75-60, 80-41, 84-99 . . . 
(Noyes and Cottle, Zeit. /. phys. Chem., 27, 578, 1898.) 

(7) Deduce the order of the reaction, 

6FeCl. 2 + KCIO, + 6HCI = 6FeCl 5 + KCl + S^HO, 

from the following data : O'l equivalents * of ferrous chloride, potassium 
chlorate and of hydrochloric acid are taken, then, if x denotes the quantity 
of FeCl 2 transformed in the time t, when 

t = 5, 15, 35, 60, 170 . . . ; 

x x 10 = 4-8, 12-2, 23-8, 32-9, 52-5 . . . 

Ansr. Third order.since k only varies between 0-99 and 1-04 when dxjdt=k(a - .*)'. 
(Hood, Phil. Mag. [5], 6, 371, 1878 ; 8, 121, 1879 ; 20, 323, 1885 ; Noyes and 
Wason, Zeit. f. phys. Chem., 22, 210, 1897.) 

(8) The following observations were made on the reaction : 
CsH 5 . SO Z . OC 2 H 5 + CH ?j . OH = CH 5 OC^H- + C 6 H- a . S0 2 . OH, 

t = 5, 10, 15, 25 . . . ; 

x = 23-1, 41-3, 55-0, 74-0 . . . 

What order of reaction gives a fairly constant value for k? (Sagrebin, Zeit. /'. 
phys. Chem., 34, 149, 1900.) 

* Note the distinction between "equivalent" and "molecular" amounts. 



$s.i. TIM-: IMK<;i;.\L CALCULUS. 'Jo:> 

(9) Schwicker (Zcit. f. /;////*. (.'//<;;/., 16, 303, 1895) has made two series of 
experiments on the action of iodine on potash. In the first series he used an 
excess of potash and found that when t = 2, a = 10'7 of iodine and when 

t =' 6, 11, 28, 38, 08 . . . ; 

x = 2-10, 2-30, 5-68, 6'50, 7'86 . . . 

Hence show that reaction between iodine and excess of potash is of the second 
order. In a second series of experiments, an excess of iodine was used. 
a = 7'23 after the elapse of two minutes, and subsequently, when 
t = 4, 8, 13, 36 . . . ; 

x = 3-43, 4-33, 4-88, 5-40 ... 

Show that the reaction is probably of the third order. These results led 
Schwicker to the equations, 

7 2 + 2KOH = KIO + KI + H. 2 O ; SKIO = 2KI + KIO 3 . 

(10) It is intended to investigate the rate of combination of hydrogen and 
oxygen gases at 440. Assuming that the reaction is of the third order, 
arrange velocity equations for the following mixtures : 

(a) 2# 2 + O a ; (b) 4# 2 + O 2 ; (c) 2H 2 + 20 2 . 

For (a) use (7), since a = b = c ; for (b) use (9) substituting a = 1, 6 = c = 2, 
and for (c) substitute a 2, b = c 1 in (9). Then arrange the results for the 
indirect determination of x, by measuring the pressure of the mixed gases as 
example (1). (Compare Bodenstein, Zeit. f. phijs. Cliem., 29, 664, 1899). 



89. Chemical Equilibrium Incomplete or Reversible 
Reactions. 

Whether equivalent proportions of sodium nitrate and potas- 
sium chloride, or of sodium chloride and potassium nitrate, are 
mixed together in aqueous solution at constant temperature, each 
solution will, after the elapse of a certain time, contain these four 
salts distributed in the same proportions. Let m and n be positive 
integers, then 

(m + n)NaN0. 3 + (m + n)KCl = mNaCl + mKNO s + nNaNO s + nKCl ; 
(m + n)NaCl + (m + n)KN0 3 = mNaCl + mKNO., + nNaN0 3 + nKCL 
This is more concisely written, 

NaCl + KN0 3 ^ NaNOz + KCL 

The phenomenon is explained by assuming that the products of 
the reaction interact to reform the original components simul- 
taneously with the direct reaction. That is to say, two inde- 
pendent and antagonistic changes take place simultaneously in 
the same reacting system. When the speeds of the two opposing 
reactions are perfectly balanced, the system appears to be in a 
stationary state of equilibrium. This is an illustration of the 
principle of the coexistence of different actions, page 52. 



204 HIGHER MATHEMATICS. 89. 

The special case of Wilhelmy's law dealing with these " in- 
complete " or reversible reactions is known as Guldberg and 
Waage's law.* 

Consider a system containing two reacting substances A } and 
A 2 such that 

Let a x and <z 2 be the respective concentrations of A l and A 2 . Let 
x of A l be transformed in the time t, then by Wilhelmy's law 



Further, let x' of A% be transformed in the time t. The rate of 
transformation of A 2 to A^ is then 

'bx'/'bt = k 2 (a 2 - x'). 

But for the mutual transformation of a? of A l to A 2 and x of A z 
to ^4 19 we must have, for equilibrium, 

x = - x and dx - dx' ; 
or, 'bx/'bi = - k 2 (a 2 + x). 

The net, or total velocity of the reaction is obviously the algebraic 
sum of these " partial " velocities, or 

dx/dt = ^(fl^ - x) - Jc 2 (a 2 + x). . (1) 

It is usual to write K = kjk^ When the system has attained the 
stationary state dx/dt = 0. (Why ?) And 

K = (a, + x}l(a l -x), . . . (2) 

where x is to be determined by chemical analysis, a^ is the amount 
of substance used at the beginning of the experiment, a 2 is made 
zero when t = 0. This determines K. Now integrate (1) by 
the method of partial fractions and proceed as indicated in the 
subjoined examples. 

The more important memoirs for consultation are Berthollet, Essai de 
Statique Chimie, Paris, 1801-1803, or Ostwald's Klassiker, No. 74 ; Wilhelmy, 
Pogg. Ann., 81, 413, 1850; Ostwald's Klassiker, No. 29; Berthelot and Gilles, 
Ann. de Chim. et d. Phys. [3], 65, 385, 1862 ; 66, 5, 1862 ; 68, 225, 1863 ; Har- 
court and Esson (I.e.) ; Guldberg and Waage, Journ. filr praktisclie Chemie 
[2], 19, 69, 1879 ; Ostwald's Klassiker, No. 104. 

EXAMPLES. (1) In aqueous solution y-oxybutyric acid is converted into 
7-butyrolactone and 7-butyrolactone is transformed into 7-oxybutyric acid 
according to the equation, 

. CH 2 . CH 2 . COOH^. CH 2 . CH^. CH. Z . CO + H Z O. 

I 



* It is, of course, just as easy to consider irreversible reactions as special types 
of Guldberg and Waage's law by supposing the velocity of tliQ reverse action, zero. 
I have followed the subject historically. 



89. THE INTEGRAL CALCULI - -jor, 

Use the preceding notation and show that the velocity of formation of the 
lactone is, 

dxldt = k^a, - x) - fc 2 ( 2 + x), . . . . (3) 

and K = kjk^ = (a z + x)l(a l - x) ..... (4) 

Now integrate (3) by the method of partial fractions. Evaluate the integra- 
tion constant for x = when t = and show that 

K ~ ( - (5) 



Henry (Zeit. f. phys. Chem., 10, 116, 1892) worked with a^ = 18-23, a^ = ; 
analysis showed that when dxldt = 0, x = 13-28 ; % - x = 4-95 ; a 2 + x = 13-28 ; 
K = 2-68. Substitute these values in (5) ; reduce the equation to its lowest 
terms and verify the constancy of the resulting expression when the following 
pairs of experimental values are substituted for x and t, 

t = 21, 50, 65, 80, 160 . . . ; 

x = 2-39, 4-98, 6-07 7'14, 10-28 . . . 

(2) A more complicated example than the preceding reaction of the first 
order occurs during the esterification of alcohol by acetic acid. 

CH, . COOH + C^H 5 . OH^ CH 3 . COOC 2 H 5 + H. OH, 
a reaction of the second order. 

Let a lt &! denote the initial concentrations of the acetic acid and alcohol 
respectively, 2 , 6 2 of ethyl acetate and water. Show that, 

dxldt = A^K - x) (b, - x) - k 2 (a. 2 + x) (6 2 + x). . . (6) 
Here, as elsewhere, the calculation is greatly simplified by taking gram mole- 
cules such that rtj = !,&! = 1, 2 = 0, 6 2 = 0. Equation (6) thus reduces to 

dxjdt = k,(l - x)* - k 2 x* ..... (7) 

For the sake of brevity, write kiK^ - k 2 ) = m and let o, be the roots of the 
equation x - 2mx + m 0. Show that (7) may be written 

dxf(x - a) (x - 0) = (&J - k 2 )dt. 

Integrate for x = when t = 0, in the usual way. Show that since 
a = m + x'w 2 - m and = m - \im 2 - m, page 387, 
1 , (m - \/m 2 - m)(m + \/w 2 - m - x) 

g = 2 * 



The value of K is determined as before. Since 

m = fc^ - fr 2 ); m = 1/(1 
Berthelot and Gilles' experiments show that for the above reaction, 

fcj/fcj = 4 ; m = I ; Vm 2 - m = | ; 

^(fej - /c 2 ) = 0-00575; or, using common logs., K^ - fc 2 ) = 0*0025. The cor- 
responding values of x and t were, 

t = 64, 103, 137, 167 . . . ; 

x = 0-250, 0-345, 0-421, 0-474 . . . ; 

constant = 0-0023, 0-0022, 0-0020, 0-0021 . . . 

Verify this last line. For^smaller values of t, side reactions are supposed to 
disturb the normal reaction, because the value of the constant deviates some- 
what from the regularity just found. 

(3) Let one gram molecule of hydriodio acid in a v litre vessel be heated, 
decomposition takes place according to the equation : 



206 HIGHER MATHEMATICS. $ 89. 

Hence show that for equilibrium, 



and that (1 - 2x)jv is the concentration of the undissociated acid. Put 
A-j/fej = K and verify the following deductions, 

- 2ar) + x 



_ 
' 8 



_ 

- x ~ v* 



Since, when tf = 0, a- = 0, C = 0. . Bodenstein (Zeit. f. phys. Clwm., 13, 56, 
1894; 22, 1, 1897) found K, at 440 = 0-02, hence \'K = 0-141, 



= constant, 



provided the volume remains constant. The corresponding values of x and are 
to be found by experiment. E.g., when ^ = 15, x = '0378, constant = 0-0171 ; 
and when ^ = 60, x = 0-0950, constant -0-0173, etc. 

(4) The "active mass" of a solid is independent of its quantity. Hence, 
if c is any arbitrary constant, show that for 

K 2 CO 3 + BaSOj, <^ K 2 S0 4 + BaCO 3 , Kc = <r/(l - x) ; 

CaCl 2 + # 2 C 2 4 2 % HCl + OC 2 O 4 , Kc = x/(l - a-) 2 ; 

CaCO s ^ CaO + CO 2 , Kc = >, 

where jo denotes the pressure of the gas. The first reactions take place in 
solution, the latter in a closed vessel. Write down the velocity equations 
before equilibrium is set up and arrange the results in a form suitable for 
experimental verification. 

(5) Prove that the velocity equation of a complete reaction of the first 
order, A^ A 2 , has the same general form as that of a reversible reaction, 
A- ^ A%, of the same order when the concentration of the substances is re- 
ferred to the point of equilibrium instead of to the original mass. 

Let | denote the value of x at the point of equilibrium, then, 

dxjdt = &!(], - a:) - k^x, becomes, dx/dt = k^ct^ - |) - fr 2 . 
Substitute for k 2 its value k^ct^ - )/ when dx/dt = 0, 

.-. dxfdt = Vi(l - )/{; or . d-xldt = fc({ - x), . . (10) 
where the meanings of a, k, k will be obvious. 

(6) Show that k is the same whether the experiment is made with the 
substance A lt or A 2 . 

It has just been shown that starting with A^ k ^V'i/1' starting with A 2 , 
it is evident that there is % - | of A 2 will exist at the point of equilibrium. 
Hence show 



therefore, as before, k^K^ - |) = 

Integrate the second of equations (9) between the limits t = and t = /, x = x 

and x = x x , thus, 

{log(( - a:,) - log({ - ar a )}/^ - constant. 

Show, from the following observations by Waddell (Journal of Physical 
Cliemistry, 2, 525, 1898), on the reciprocal conversion of ammonium thio- 
cyanate into thiourea, that it makes no difference to the value of k, in (10), 
whether thiourea, or thiocyanate is used at the start. 



THE INTEGRAL CALCULUS. 



207 



First, the conversion of thiocyanate into thiourea, $ 21-2/ of thiocyanate, 
t = 0, 19, 38, 48, 60, . . . ; 

x = 2-0, 6-9, 10-4, 12-3, 13'5, . . . 
Second, the conversion of thiourea into thiocyanate, = what ? 
t = 0, 38, 53, 68, 90, . . . ; 

x = 81-1, 51-5, 54-4, 56-3, 65'0, . . . 

Memoirs by Walker and Hambly, Journ. Chem. Soc., 67, 746, 1895 ; Walker 
and Appleyard, ib., 69, 193, 1896; Waddell, I.e., 3, 41, 1899, and Kistiakovvsky, 
Zfif. f. phyx. Client., 27, 258, 1898, may be consulted with reference to this 
nu'thod of developing equilibrium equations. 



90. Fractional Precipitation. 

If to a solution of a mixture of two salts, A and B, a third 
substance C, is added, in an amount insufficient to precipitate all 
A and B in the solution, more of one salt will be precipitated, as 
a rule, than the other. By redissolving the mixed precipitate and 
again partially precipitating the salts, we can, by many repetitions 
of the process, effect fairly good separations of substances otherwise 
intractable to any known process of separation. 

Since Mosander thus fractioned the gadolinite earths in 1843 
(Hood, Phil. Mag., [5], 21, 119, 1886), the method has been ex- 
tensively employed by Crookes in some fine work on the yttria 
and other earths. The recent separations of polonium, radium 
and other curiosities has attracted some attention to the process. 
The " mathematics " of the reactions follows directly from the law 
of mass action. 

Let only sufficient C be added to partially precipitate A and B 
and let the solution originally contain a of the salt A, b of the salt 
B. Let x and y denote the amounts of A and B precipitated at 
the end of a certain time t, then a - x and b - x will represent the 
amounts of A and B respectively remaining in the solution. The 
rates of precipitation are, therefore, 

dx/dt = k(a - x) (c - z); dy/dt = k'(b - y) (c - z), 
where c - z denotes the amount of C remaining in the solution at 
the end of a certain time t. 

.'. dx/dt :dy/dt = k(a - x):k'(b - y), 

or, V ( a ~ x ) = k\ \ ~ y \ 

J a - x J b - y 

or, k'\og(a - x) = k log(6 - y) + log C', 

where log C' is the integration constant. 



208 HIGHER MATHEMATICS. 91. 

To find C' put x = and y = 0, then 

log a*' = logC'6*, or, C' = a k '{b k . 

Therefore, 1 = log(a - x)/a (1) 

k' log(b - y)/b 

The ratio (a - x)/a measures the amount of salt remaining in the 
solution, after x of it has been precipitated. The less this ratio, 
the greater the amount of salt A in the precipitate. The same 
thing may be said of the ratio (b - y)/b in connection with the 
salt B. 

The more k exceeds k', the less will A tend to accumulate in 
the precipitate and, the more k' exceeds k, the more will A tend ta 
accumulate in the precipitate. If the ratio k/k' is nearly unity, the 
process of fractional precipitation will be a very long one. In the 
limiting case, when k = k', or, k/k' = 1, the ratio of A to B in the 
mixed precipitate will be the same as in the solution. In such a 
case, the complex nature of the " earth " could never be detected 
by fractional precipitation. 

The application to gravimetric analysis is obvious. 

91. The Isothermal Expansion of Gases. 

To find the work done during the isothermal* expansion of a gas. 
Case i. The gas obeys Boyle's law, 

pv = constant say, c. 

On page 182 it was shown that the work done when a gas ex- 
pands against any external pressure is represented by the product 
of the pressure into the change of volume. The work performed 
during any small change of volume, is 

dW = p.dv ..... (1) 
But by Boyle's law, 

p=f(v) = dv. . (2} 

Substitute this value of p in (1), and 

.-. dW = c . dv/v. 
If the gas expands from a volume v 1 to a new volume i? 2 , 



c>gt?+ C; 
* K 

or, TF=clog- 2 . . . (3) 



* " Isothermal " means " at a constant temperature," as pointed out on page 90. 



91. THE INTEGRAL CALCULUS. 209 

From (2), v l = c/p l and v 2 = c/p 2 , hence, 

TP = clog&. . (4) 

Pz 
Equations (3) and (4) play a most important part in the theory 

of gases, in thermodynamics and in the theory of solutions. 

The value of c is equal to the product of the initial volume (v ) 
and pressure (p ) of the gas, 

.-. W = 2-3026p Vog 10 i ; 



Pi 

See page 520, for a numerical example. 

Case ii. The gas obeys van der Waals' law, 

(p + ~ V - b) = constant, say, c'. 



As an exercise prove that 

(5) 



This equation has occupied a prominent place in the develop- 
ment of van der Waals' theories of the constitution of gases and 
liquids. 

Case iii. The gas dissociates during expansion. (After Nernst 
and Schonflies.) 

By Guldberg and Waage's law, in the reaction : 



for equilibrium, 

T J. - x xx 
K. = -.-. 

V V V 

where (1 - x)jv represents the concentration of the undissociated 
nitrogen peroxide. 

The relation between the volume and degree of dissociation is, 
therefore, 

Kv = z 2 /(l -x). ... (6) 

where x the fraction of unit mass of gas dissociated. 

If n represents the original number of molecules (1 - x)n will 
represent the number of undissociated molecules and 2xn the num- 
ber of dissociated molecules. If the relation pv = c, does not vary 
during the expansion, the pressure will be proportional to the num- 
ber of molecules actually present, that is to say, 
n : {(1 - x)n + 2xn] = 1 : 1 + x. 
O 



210 HIGHER MATHEMATICS. 91. 

The actual pressure of the gas is, therefore, 

p = (1 + x)p, 
and the work done is, therefore, 

dW = p . dv = (1 + x)p .dv=p.dv + xp. dv. 
But dW l = p . dv &ud dW 2 = xp.dv, . . (7) 

and W=W 1 + W Z ..... (8) 

From Boyle's law, p = c/v, and (6), 

. c c 



Substitute this value of p in (7). Differentiate (6) and substitute 
the value of dv so obtained in our last result. Simplify and 
dW 2 = c(2 - x)dxl(l -x) = c{l + 1/(1 - x)}dx. 

Integrate TF 2 = cPYl + l \dx, 

Jx 2 \ 1 x/ 

where x t and x 2 denote the values of x corresponding to v l and v%. 

.-. W 2 = c{(x, - Xl ) - log(l - OJ 2 )/(1 - x,)}. 
Eind dW l in a similar way from (7). 

W 1 = clogC^i)- 
.-. W = c(log^ + x, - x, - log J_l|i). (9) 

It follows from (6), that 



, and v<> = 

Substitute these values of v in (9) 

~, /i ~, \\ 

. (10) 



EXAMPLES. (1) Find the work done during the isothermal expansion of 
dissociating ammonium carbamate, supposed gaseous. 
NH 2 COONH 4 ^2NH S + C0 2 . 

(2) In calculating the work done during the isothermal expansion of 
dissociating hydrogen iodide, 



does it make any difference whether the hydrogen iodide dissociates or not ? 

(3) If the force of attraction (/) between two molecules of a gas, varies 
inversely as the fourth power of the distance (r) between them, show that the 
work (W) done against molecular attractive forces when a gas expands into 
a vacuum, is proportional to the difference between the initial and final 
pressures of the gas. That is, 

W^Afa-pJ, ..... (11) 
where A is the variation constant of 189. By hypothesis, 
/=a/r*: and, dW = f.dr, 



92. THE INTEGRAL CALCULUS. 211 

where a is another variation constant. (See 79.) Hence, 



But r is linear, therefore, the volume of the gas will vary as r 3 . Hence, 
v = &r", where 6 is again constant. 



But by Boyle's law, pv = constant, say, = c. Hence it follows, 

W '= A(p r - PZ), if A = ab/3c = constant. 

(4) // the work done against molecular attractive forces when a gas 
expands into a vacuum, is 



v 9 , 

where a is constant ; v lt v 2 , refer to the initial and final volumes of the gas, 
show that "any two molecules of a gas will attract one another with a force 
inversely proportional to the fourth power of the distance between them ".* 



92. The Adiabatic- Expansion of Gases. 

In one of the examples appended to 26, we obtained the 
expression, 



As pointed out on page 29, we may, without altering the value 
of the expression, multiply and divide each term within the brackets 
byd0. .Thus, 



But (bQ/~dO) p is the amount of heat added to the substance at a 
constant pressure for a small change of temperature ; this is none 
other than the specific heat at constant pressure, usually written 
C p . Similarly (&Ql"bO), is the specific heat at constant volume, 
written C v . 

P . . . (3) 

This equation tells that when a certain quantity of heat is added 
to a substance, one part is spent in raising the temperature while 
the volume changes under constant pressure, and the other part is 

* For the meaning of a/v 2 , see van der Waals' equation. 

f The substance is supposed to be in such a condition that no heat can enter or leave 
the body during the expansion. The temperature, therefore, may change during the 
operation. 



212 HIGHER MATHEMATICS. 92. 

spent in raising the temperature while the pressure changes under 
constant volume. 

For an ideal gas obeying Boyle's law, 

pv = Re. 

.-. v/B = (Wftp); p/B = (Wftv). 
Substitute these values in (3), 

.-. dQ = C p p . dv/R + C v v . dp/R. 
Divide through by = pv/R, and, 

^ = C*L + a.*. (4) 

v p 

By definition, an adiabatic change takes place when the system 
neither gains nor loses heat, that is to say, dQ = 0. 

The ratio of the two specific heats C P /C V is a constant, usually 
written y. 

C n dv dp n fdv Cdv 

.-. - . -- |_^c=0; or, yl + I-J: = constant. 
C v v p J v J p 

or, ylogv + logp = constant; or, log^ Y + logp = constant, 

.-. log(pv y ) = constant; or, pv y = constant. . (5) 
A most important relation in the theory of thermodynamics. 

By integrating between the limits p v p 2 and v lt v 2 in the 
above equation, we could have eliminated the constant and ob- 
tained 

a-^Y' .... (6) 

PI W 

a useful form of (5). 

Substituting v l = O l R/p l and v z = G^RIp^ in (6), 



PI PI 

and from (6) ?i = l ..... (8) 



Equation (6), in words, states that the- adiabatic pressure of a 
gas varies inversely as the yth power of the volume. Equation 
(8) affirms that for adiabatic changes, the absolute temperature of 
a gas varies inversely as the (y - l)th power of the volume. Two 
well-known thermodynamic laws. 

To find the work performed when a gas is compressed under 
adiabatic conditions. 

From (5), if we write the constant c', 
p = c'/v y . 






THE INTEGRAL CALCULUS. 



213 



Since the work done when the volume of a gas is compressed from 
v l to v 2 is (page 182), 






f"a fa dv 

p.dv--\ c' ; 
J' vy 





- (y - 1) = T^Tl^l - ^T 

From (5), c' = p^oj = p 2 vj. We may, therefore, represent this 
relation in another form, viz. : 




and p 2 v. 2 = #0 2 , are the isothermal equations for 



If 



and 2 , we may write, 



which states in words, that the work required to compress a mass 
of gas adiabatically while the temperature changes from 6-f to # 2 , 
will be independent of the initial pressure and volume of the gas. 
In other words, the work done by a perfect gas in passing along 
an adiabatic curve, from one isothermal to another, is constant 
(see page 92), and independent of the path. 

EXAMPLES. (1) From (5), show that p^v-^ = p^v^ and hence deduce the 
formula, 

7 = (iog.Po - log^i)/(logJP 2 - lo g^i). ( 12 ) 
used by Clement and Desormes in their determinations of the ratio of the 
specific heats of some of the gases (Journ. de Physique, 89, 333, 1819). The 
experimental details are given in most textbooks. Here it is only necessary 
to know that p l v l = p^v under the conditions of the experiment. The numeri- 
cal values of p , p lt p 2 , are determined by experiment. 

(2) To continue illustration 3, 18, page 44. We have assumed Boyle's 
law pp Q = pop. This is only true under isothermal conditions. For a more 
correct result, use (5) above. Write the constant c. For a constant mass (m) 
of gas, m = pv, hence show that for adiabatic conditions, 



Hence deduce the more correct form of Halley's law : 






for the pressure (p) of the atmosphere at a height h above sea-level. Atmos- 
pheric pressure at sea-level = p . 

(3) From the preceding example proceed to show that the rate of diminu- 



214 HIGHER MATHEMATICS. 93. 

tion of temperature (6) is constant per unit distance (h) ascent. In other 
words, prove and interpret 

-0 = ^-.^ U (15) 

R y 

(4) Lummer and Pringsheim have used the last of equations (7), for 
evaluating 7 by allowing a gas at pressure p- to expand suddenly to another 
pressure p 2 and measuring the instantaneous rise of temperature B 1 to 2 . 
Hence, given the numerical values of p lt p 2 , 6 V 2 , how would you calculate 
the numerical value of 7? Ansr. 7 = lo&(piJPiM$o&(pilPz) - logi^/flaK- 

(5) To continue the discussion at the end of 26, Examples (4) to (8). 
Suppose the gas obeys van der Waals' law : 

- b) = RB, . . . . (16) 



where R, a, b, are known constants. The first law of thermodynamics may 
be written 

dQ = C v . de + (p + alv*)dv, .... (17) 

where the specific heat at constant volume has been assumed constant. To 
find a value for C p , the specific heat at constant pressure. Expand (16). Differ- 
entiate the result. Cancel the term 2a& . dvjv* as a very small order of magni- 
tude ( 4). Solve the result for dv. Multiply through with p + ajv" 2 . Since 
a/v 2 is very small, show that the fraction (p + a/v 2 )l(p - ajv 2 ) is very nearly 
1 + 2a/jiw 2 (pages 8 and 224). Substitute the last result in (17), and 



Obviously the coefficient of d6 is equivalent to (dQ/?)6)p, i.e., to C p ; while the 
coefficient of dp is (dQj'dp)0. By hypothesis C v is constant, 



For ideal gases a = 0, and we get Mayer's equation, 26. 
For. Air. Hydrogen. 

l lc , : : : ST 18 ST 8 " 

7 (calculated) . 1-40225 1-40007 1-2907 From (18) ; 

7 (observed) 1-403 1-4017 - 1-2911 / Mean of d f ta in M ^ er ' s 

{Kinetic Theory of Gases. 

(6) Show van der Waals' equation for adiabatic conditions is 

-b)y = Be ..... (19) 



93. The Influence of Temperature on Chemical and 
Physical Changes van't Hoff s Formula. 

In example 7, page 61, we have obtained the formula, 

(S).-(l). ..... * 

by a simple process of mathematical reasoning. The physical 
signification of this formula is that the change in the quantity 



$ <:;. THE INTEGRAL CALCULUS. 215 

of heat communicated to any substance per unit change of volume 
at constant temperature, is equal to the product of the absolute 
temperature into the change of pressure per unit change of tem- 
perature at constant volume. 

Suppose that 1 - x grams of one system A is in equilibrium 
with x grams of another system B. Let v denote the total volume 
and the temperature of the two systems. Equation (1) shows 
that (bQIDv)e is the heat absorbed when the very large volume of 
system A is increased by unity at constant temperature 0, less the 
work done during expansion. Suppose that during this change of 
volume, a certain quantity (tto/dv)* of system B is formed, then, if 
q be the amount of heat absorbed when unit quantity of the first 
system is converted into the second, the quantity of heat absorbed 
during this transformation is q(bxfbv)t. q is really the molecular 
heat of the reaction. 

The work done during this change of volume is p . dv ; but dv 
is unity, hence the external work of expansion is p. Under these 
circumstances, 

/<te\ fiQ\ fip\ 0^p - p^O 

U). = (*).-* ' (5). - p ' -*&- w 

from (1). Now multiply and divide the numerator by 2 (see 
integrating factors, pages 58 and 120). 

.- 



If, now, ?i x molecules of the system A and n. 2 molecules of the sys- 
tem B take part in the reaction, we must write, instead of pv = RO, 

pv = BOfo^l - x) + n 2 x] ; or, p/e = E\n^ + (n. 2 - n^x^v. 
(The reason for this is well worth puzzling out.) 

R . 



Substitute this result in equation (9) and we obtain 

PR. 



By Guldberg and Waage's statement of the .mass law, 



.-. log K + (n. 2 - n^ log v = n. 2 log x - w x log (1 - x). 
Differentiate this last expression with respect to 0, at constant 
volume and with respect to v, at constant temperature, 

n. - H 



( 
\ 



1 - x x 1 - x 



216 HIGHER MATHEMATICS. 93. 

Introduce these values in (4) and reduce the result to its simplest 
terms, thus, 

/x 

* 



This fundamental relation expresses the change of the equilibrium 
constant K with temperature at constant volume in terms of the 
molecular heat of the reaction. 

Equation (5), first deduced by van't Hoff, has led to some of 
the most important results of physical chemistry. 

Since E and are positive, K and q must always have the 
same sign. Hence van't Hoff's principle of mobile equilibrium 
follows directly, viz. : 

If the reaction absorbs heat, it advances with rise of tempera- 
ture ; if the reaction evolves heat it retrogrades with rise of tem- 
perature ; and if the reaction neither absorbs nor evolves heat, the 
state of equilibrium is stationary with rise of temperature. 

According to the particular nature of the systems considered q 
may represent the so-called heat of sublimation, heat of vaporiza- 
tion, heat of solution, heat of dissociation, or the thermal value of 
strictly chemical reactions when certain simple modifications are 
made in the interpretation of the " concentration " K. 

If, at temperature O l and 2 , K becomes jfiTj and K 2 , we get, by 
the integration of (5), 



The thermal values of the different molecular changes, calculated 
by means of this equation, are in close agreement with experiment. 
For instance : 





q in calories. 




Calculated. 


Observed. 


Heat of vaporization of water 


10100 


10296 


Heat of solution of benzoic acid in water 


6700 


6500 


Heat of sublimation of NH 4 SH . 


21550 


21640 


Heat of combination of BaCl^ + 2H 2 O 


3815 


3830 


Heat of dissociation of N 2 4 


12900 


12500 


Heat of precipitation of AgCl 


15992 


15850 



A sufficiently varied assortment to show the profound nature of 
the relation symbolised by equations (5) and (6) (see van't Hoff's 
Chemical Dynamics (Ewan's translation)). 



$ <:;. THE INTEGRAL CALCULUS. 217 

NUMERICAL EXAMPLE. Calculate the heat of solution of mercuric chloride 
from the change of solubility with change of temperature. If c lt % denote the 
solubilities corresponding to the respective absolute temperatures 0, and 2 , 

Cj = 6-57 when ^ = 273 + 10 ; c 2 = 11-84 when 2 = 273 + 50. 
Since the solubility of a salt in a given solvent is constant at any fixed tem- 
perature, we may write c in place of the equilibrium constant K. From (6), 
therefore, 

qfl 1\. . . 11-84 qf I 1 \ 
U " ej ' " 10g fr57 = 21,283 ~ 323> 
.-. q = log 1-8 x 45,704-5 = 2,700 (nearly) ; 
q (observed) = 3,000 (nearly). 

Use the Table of Natural Logarithms, Chapter XIII., for the calculation. 

Le Chatelier has reversed the above calculations, and, as the 
result of more extended investigations, he has enunciated the im- 
portant generalisation : " any change in the factors of equilibrium 
from outside, is followed by a reversed change within the system". 
This rule, known as Le Chatelier s theorem, enables the chemist to 
foresee the influence of pressure and other agents on physical and 
chemical equilibria. 

For further light on this important subject, consult Le Chatelier's Les 
Equilibres Chimiques, 1888 ; Zeit. f. phys. CJiem., 9, 335, 1892 ; Bancroft's 
The PJiase Rule, 1897. 

The beginner will find it worth while to write out the leading assumptions 
introduced as premises in deducing van't Hoff's formula. 



218 



CHAPTEE V. 
INFINITE SERIES AND THEIR USES. 

" In abstract mathematical theorems, the approximation to truth is- 
perfect. ... In physical science, on the contrary, we treat of 
the least quantities which are perceptible." W. STANLEY JEVONS, 

94. What is an Infinite Series? 



MARK off a distance AB of unit length. Bisect AB at O v bisect 
O^B at 2 , 2 B at 8 , etc. 

A O l 2 3 4 B. 

By continuing this operation, we can approach as near to B as we 
please. In other words, if we take a sufficient number of terms 
of the series, 

A0 l + O^ + 2 3 + . . . , 

we shall obtain a result differing from AB by as small a quantity 
as ever we please. 

This is the geometrical meaning of the infinite series of terms, 
1 = 4 + (i) 2 + (I) 3 + (i) 4 + ... to infinity. . (1) 
Such an expression, in which the successive terms are related 
according to a known law, is called a series. 

When the sum of an infinite series approaches closer and closer 
to some definite finite value, as the number of terms is increased 
without limit, the series is said to be a convergent series. The 
sum of a convergent series is the " limiting value " of 6. On 
the contrary, if the sum of an infinite series obtained by taking a 
sufficient number of terms can be made greater than any finite 
quantity, however large, the series is said to be a divergent series. 
For example, 

l + 2 + 3 + 4+...to infinity. . . (2) 
Divergent series are not much used in physical work, while con- 
verging series are very frequently employed.* 

* A prize was offered in France some time back for the best essay ou the use of 
diverging series in physical mathematics. 



S 114. INFINITE SKRIKS AM) THEIR USES. _'!' 

Several tests for discriminating between convergent and diver- 
gent series are described in the regular textbooks on algebra. To 
simplify matters, I shall assume the series discussed in this work 
satisfy the tests of convergency. It is necessary to bear this in 
mind, otherwise we may be led to absurd conclusions. 

Let S denote the limiting value or sum of the converging 
series. 

S = a + ar + ar 2 + . . . + ar" + ar n+l + ... ad inf. (3) 
Cut off the series at some assigned term, say the rath, i.e., all terms 
after ar' 1 ~ l are suppressed. Let s tl denote the sum of the n terms 
retained, - the sum of the suppressed terms. Then, 

s u = a + ar + ar' 2 + . . . + ar"" 1 . . . (4) 
Multiply through by r, 

rs n = ar + ar' 2 + ar* + . . . + ar". 
Subtract the last expression from (4), 

*(! - r) = a(l - r") ; or, s n = hj-^X (5) 
Obviously we can write series (3), in the form, 

8m+ : v n (6) 

The error which results when the first n terms are taken to repre- 
sent the series, is given by the expression 

<r w = S - s,, 

This error can be made to vanish by taking an infinitely great 
number of terms, or, 

Lt n = : ,<r, t = 0. 

1 - r n a ar' 1 

But, .. - *- J j-^ - j. 

When n is made infinitely great, the last term vanishes, 

,.*...-, -a 

The sum of the infinite series of terms (3), is, therefore, given 
by the expression 



Series (3) is generally called a geometrical series. 

To determine the magnitude of the error introduced when only 
a finite number of terms of an infinite series is taken. Take the 
infinite number of terms, 

S = l = 1 + r + r* + . . + r"- 1 + (8) 

I - r I - r 



220 HIGHER MATHEMATICS. $ 95. 

The error introduced into the sum S, by the omission of all terms 
after the nth, is, therefore, 

n-n 

r.- - 0) 



When r is positive, a- n is positive, and the result is a little too 
small ; but if r is negative 



which means that if all terms after the nth are omitted, the sum 
obtained will be too great or too small, according as n is odd or 
even. 

EXAMPLES. (1) Suppose that the electrical conductivity of an organic 
acid at different concentrations has to be measured and that the first 
measurement is made on 50 c.c. of solution of concentration c. 25 c.c. of 
this solution are then removed and 25 c.c. of distilled water added instead. 
This is repeated five more times. What is the then concentration of the acid 
in the electrolytic cell ? 

Obviously we are required to find the 7th term in the series 

c{i + i + a) 2 + (i) 3 + ...}, 

where the nth term is c(\} n - l . Ansr. () 6 c. 

(2) A precipitate at the bottom of a beaker containing Fc.c. of mother 
liquid is to be washed by decantation, i.e., by repeatedly filling the beaker up 
to say the Fc.c. mark with distilled water and emptying. Suppose that the 
precipitate retains v c.c. of the liquid in the beaker at each decantation, what 
will be the percentage volume of mother liquor about the precipitate after 
the nth emptying, assuming that the volume of the precipitate is negligibly 
small ? Ansr. lOOfa/ F) M * l . 

Hint. The solution in the beaker, after the first filling, has vjV c.c. of 
mother liquid. On emptying, v of this v/V c.c. is retained by the precipitate. 
On refilling, the solution in the beaker has (v 2 /F)/Fof mother liquor, and so 
we build up the series, 



95. Soret's Diffusion Experiments. 

These experiments will serve to illustrate the use that may be 
made of a geometrical series in the study of natural phenomena. 

The density of a gas may be determined by comparing its rate 
of diffusion with that of another gas of known density. If r v r 2 
be the rates of diffusion of two gases of known densities p 1 and p., 
respectively, then by Graham s law, 

r i Vft = r -2 Vp7 ... (1) 



55 ?:,. INFINITE SERIES AND THEIR USES. 221 

The method is particularly useful for finding the density of 
such a gas as ozone, which cannot be prepared free from admixed 
oxygen. Soret based his classical method for finding the density 
of this gas on the following procedure (Ann. d. Chim. et d. Phys. t 
[4], 7, 113, 1866 ; 13, 257, 1868). 

A vessel A, containing v volumes of ozone mixed with oxygen, 
was placed in communication with another vessel B, containing 
oxygen only, for a definite time t. Soret found that the volume 
(v) of ozone diffusing from A to B was proportional to the differ- 
ence in the quantity of ozone contained in the two vessels at the 
commencement of any interval of time. By Graham's law this 
quantity is also inversely proportional to the square root of its 
density. 

If the vessel A, originally containing V Q volumes of ozone, loses 
v volumes, the amount dv which diffuses in the next interval of 
time dt, will be proportional to the difference in the volumes of 
ozone contained in the two vessels, that is to say, (v - v) - v, 
hence, 

dv = -^(r - 2v)dt, ... (2) 

where a is a constant depending on the nature of the apparatus 
used in the experiment. 

At the commencement of the first interval of time B contained 
no ozone, therefore, if v 1 denotes the quantity of ozone in B at the 
end of the first interval of time, 

v i = -*T*J* ; .... (3) 
vp 

at the end of the second interval, 

v 2 = v l + v^l - 2v!/v Q ) ; 
at the end of the third interval, 

v 3 = v l + t?j(l - 2V ) + v i( l - 2^/t; ) 2 ; 
and at the end of the nth interval, 

v,, = v l + v^l - Sfy/v,,) + + ^iC 1 - 2^/Vo) 1 . (4) 
The volume of ozone in the upper vessel at the end of n in- 
tervals of time dt, is the sum of the geometrical series (4) containing 
n terms. From (5), page 219, 



Thus, the volume of the gas in B, at the end of a given time, is 



222 HIGHER MATHEMATICS. $ 96. 

proportional to v alone, or, for the same gas with the same 
apparatus for the same interval of time, 
vjv = constant. 

With different gases, under the same conditions, any difference in 
the value of vJv Q must be due to the different densities of the 
gases. 

The mean of a series of experiments with chlorine (density, 
35 -5), carbon dioxide (density, 22), and ozone (density, ?), gave 

the following numbers : 

CO 2 . Ozone. C/ 2 . 

vjv . . . 0-29, 0-271, 0-227. 
Comparing chlorine with ozone, let x denote the density of 
ozone, 

x = (0-227/0-271) 2 x 35-5 = 24-9, 
which agrees with the triatomic symbol 3 . 

EXAMPLE. Show that if the time is taken infinitely long the value of 
v n ]v approaches unity. 

96. Approximate Calculation by Means of Infinite Series. 

The reader will, perhaps, have been impressed with the fre- 
quency with which experimental results are referred to a series 
formula of the type : 

y = A + Ex + Cx 2 + Dz 3 + . . ., . . (1) 
in physical or chemical textbooks.* 

The formula has no theoretical significance whatever. In the 
absence of any knowledge as to the proper mathematical expres- 
sion of the " law " connecting two variables, this formula is 
adopted in the attempt to represent the corresponding values 
of the two variables by means of a mathematical expression. 

A, B, C, . . . are constants to be determined from the ex- 
perimental data by methods to be described later on. 

There are several interesting features about formula (1). 

1. When the progress of any physical change is represented by 
the above formula, the approximation is closer to reality the greater 
the number of terms included in the calculation. This is best 
shown by an example. 

The specific gravity s of an aqueous solution of hydrogen 

* I have counted over thirty examples in the first volume of MendeleefFs The 
Principles of Chemistry and more than this number in Preston's Theory of Heat. 



55 



INFINITE SERIES AND THEIR USES. 



chloride is an unknown function of the amount of gas p per cent, 
dissolved in the water. (Unit, water at 4 = 10,000.) 

The first two columns of the following table represent cor- 
responding values of p and s, determined by Mendeleeff. It is 
desired to find a mathematical formula to represent these results 
with a fair degree of approximation, in order that we may be able 
to calculate p if we know s, or, to determine s if we know p. Let 
us suppress all but the first two terms of the above series, 

s = A + Bp, 

where A and B are constants, found, by methods to be described 
later, to be A = 9991'6, B = 4943. Now calculate s from the 
given values of p by means of the formula, 

s = 9991-6 + 49-43p, ... (2) 

and compare the results with those determined by experiment. 
See the second and third columns of the following table : 





Specific Gravity s. 


Percentage 
Composition 




Calculated. 


P- 










1st Approx. 


2nd Approx. 


5 


10242 


10239 


10240 


10 


10490 


10486 


10492 


15 


10744 


10733 


10746 


20 


11001 


10980 


11003 


25 


11266 


11227 


11263 


30 


11522 


11476 


11522 



Formula (2), therefore, might serve all that is required in, say, 
a manufacturing establishment, but, in order to represent the con- 
nection between specific gravity and percentage composition with 
a greater degree of accuracy, another term must be included in the 
calculation, thus we write 

s = A + Bp + Cp\ 

where C is found to be equivalent to 0-0571. The agreement 
between the results calculated according to the formula : 

s = 9991-6 + 49-43p + 0'0571p*, . . (3) 
and those actually found by experiment is now very close. This 
will be evident on comparing the second with the fourth columns 
of the above table. 



224 HIGHER MATHEMATICS. 96. 

The term 0*0571p 2 is to be looked upon as a correction term* 
It is very small in comparison with the preceding terms. 

If a still greater precision is required, another correction term 
must be included in the calculation, we thus obtain 
y = A + Bx + Cx 2 + Dx*. 

Such a formula was used by Thorpe and Tutton (Journ. Chem. 
Soc., 57, 559, 1890 ; Thorpe and Kucker, Phil. Trans., 166, ii., 
405, 1877), to represent the apparent expansion of phosphorous 
oxide in a glass volumeter. They referred their- results to the 
formula : 

v = l + 0-008882,40 + ( - 0-000000,13873)^ + 0'000000,0384460 3 . 
The calculated agreed very closely with the observed results, 
(Thorpe and Tutton's zero temperature was here - 27'1.) 

Hirn used yet another term, namely, 

v = A + Bo + Ce 2 + De* + Ee*, 

in his formula for the volume of water, between 100 and 200. 
Here 4 = 1, 

B = . 0-000108,67875 ; D = 0-000000,002873,0422 ; 

C = 0-000003,007365,3 ; E = - 0-000000,000006,646703,1. 
(Ann. d. Ch. et d. Ph. [4], 10, 32, 1867.) 

The logical consequence of this reasoning, is that by including 
every possible term in the approximation formula, we should get 
absolutely correct results by means of the infinite converging series : 

y = A + Bx + Cx 2 + Dx* + Ex* + Fx* + . . . + ad infin. 

It is the purpose of Maclaurin's theorem to determine values of 
A, B, C, . . . which will make this series true. 

2. The rapidity of the convergence of any series determines how 
many terms are to be included in the calculation in order to obtain 
any desired degree of approximation. 

It is obvious that the smaller the numerical value of the " cor- 
rection terms " in the preceding series, the less their influence on 
the calculated result. If each correction term is very small in 
comparison with the preceding one, very good approximations can 
be obtained by the use of comparatively simple formulae involving 
two, or, at most, three terms. On the other hand, if the number 
of correction terms is very great, the series becomes so un- 
manageable as to be practically useless. 

Equation (1) may be written in the form, 

y = A(l + bx + ex 2 + ...),. . . (4) 
where A, b, c, . . . are constants. 



$ MJ. INFINITE SERIES AND THEIR USIX 225 

As a general rule, when a substance is heated, it increases in 
volume ; its mass remains constant, the density, therefore, must 
necessarily decrease. But, 

mass = volume x density, or, m = pv. 
The volume of a substance at is given by the expression 

V = V Q (l + aO), 

where V Q represents the volume of the substance at C., a is the 
coefficient of cubical expansion. Therefore, 

P /p = v/v = v (l + aO)/v Q = 1 + aO. 

.'. p = p ft /(l + a0). 

True for solids, liquids, and gases. For simplicity, put p () = 1. By 
division, we obtain 

p = 1 - a.0 + (aO)' 2 - (a0) 3 + . . . 

For solids and some liquids a is very small in comparison with 
unity. For example, with mercury a = 0*00018. Let 6 be small 
enough 

p = I - 0-000180 + (0-000180) 2 - ... 

= 1 - 0-000180 + 0-000000,03240* 

If the result is to be accurate to the second decimal place (1 per 100), 
terms smaller than O'Ol should be neglected ; if to the third decimal 
place (1 per 1000), omit all terms smaller than O'OOl, and so on. 
It is, of course, necessary to extend the calculation a few decimal 
places beyond the required degree of approximation. How many, 
naturally depends on the rapidity of convergence of the series. 
If, therefore, we require the density of mercury correct to the 
sixth decimal place, the omission of the third term can make no 
perceptible difference to the result. See the determination of the 
numerical value of TT, page 230. 

EXAMPLES. (1) If fe denotes the height of the barometer at C. and 
h its height at 6, what terms must be included in the approximation 
formula, 

h = fe (l + 0-000160), (5) 

in order to reduce a reading at 20 to the standard temperature, correct to 
1 in 100,000? 

(2) Verify the first half-dozen approximation formulae, page 486. 

(3) In accurate weighings a correction must be made for the buoyancy of 
the air by reducing the "observed weight in air" to "weight in vacuo ".* Let 



* A difference of 45 mm. in the height of a barometer during an organic combustion 
analysis, may cause an error of 0'6 / in the determination of the (70 2 , and an error of 
0-4 / in the determination of the JI Z 0. See Crookes, "The Determination of the 
Atomic Weight of Thallium," /'hit. 7Vr<//*., 163,277, 1874. 

P 



226 HIGHER MATHEMATICS. $ 07. 

W denote the true weight of the body (in vacuo), w the observed weight in air, 
p the density of the body, ^ the density of the weights, p 2 the density of the 
air at the time of weighing. Hence show that if 



Pi 



Pi P 

or, W=w + 0-0012w(l/ P - l/ Pl ), .... (6) 

which is the standard formula for reducing weighings in air to weighings in 
vacuo. The numerical factor represents the density of moderately moist air 
at the temperature of a room under normal conditions. 

(4) If a denotes the coefficient of cubical expansion of a solid, the volume 
of a solid at any temperature 6 is, v = v (I + a0), where V Q represents the 
volume of the substance at 0. Hence show that the relation between the 
volumes, ^ and v 2 , of the solid at the respective temperatures of 0j and 2 is 

v l = v 2 (l + aQ l - a0 2 ). (7) 

Why does this formula fail for gases ? 

(5) Since 

_l_ = :L + . + + . .., 

the reciprocals of many numbers can be very easily obtained correct to many 
decimal places. Thus 

1 = 1 JL 3 9 

97 ~ 100 - 3 ~ 100 + 10,000 + 1,000,000 + 
= -01 + -0003 + -000009 + . . . 

(6) We require an accuracy of 1 per 1,000. What is the greatest value of 
x which will permit the use of the approximation formula 

(1 + x) 3 = 1 + 3x ? 

(7) From the formula 

(1 + x) n - 1 nx, 

calculate the approximate values of V99, I/ Vf-02, (1-001) 3 , vT-05, mentally. 
Note n may be positive or negative, integral or fractional. 



97. Maclaurin's Theorem. 

There are several methods for the development of functions 
in series, depending on algebraic, trignometrical, or other pro- 
cesses. The one of greatest utility is known as Taylor's theorem. 
Maclaurin's * theorem is but a special case of Taylor's. 

Maclaurin's theorem determines the law for the expansion of a 
function of a SINGLE variable in a series of ascending powers of 
that variable. 

* The name is here a historical misnomer. Taylor published his series in 1715. 
In 1717, Stirling showed that the series under consideration was a special case of 
Taylor's. Twenty-five years after this Maclaurin independently published Stirling's 
series. 



jj !>7. INFINITE SERIES AND THEIR USES. 227 

Let the variable be denoted by x, then, 

= /(*). 

Assume that f(x) can be developed in ascending powers of x, 
say, 

u =f(x) = A + Ex + to 2 + Dx* +...,* . . (1) 
where A, B, C, D . . . , are constants independent of #, but de- 
pendent on the constants contained in the original function. 

It is required to determine the value of these constants, in order 
that the above assumption may be true for all values of x. 

By successive differentiation of (1), 



_ . (2) 

dx dx 

4*.W*>, aC + a.SDx + . . .; . (3) 

dx 2 dx 

*_#"(?)- 2. 3. D+. (4) 

dx* dx 

By hypothesis, (1) is true whatever be the value of x, and, 
therefore, the constants A, B, C, D, . . . are the same whatever 
value be assigned to x. Now substitute x = in equations (2), (3), 
(4). Let v denote the value assumed by u when x = 0. Hence, 
from (1), 

v=f(0) = A, .'.A = v, . (5) 

from (2), - 



from(3), 

from w, g-rm-i.i.aD, .-^ = i. 

u / n (0) " means that /(a?) is to be differentiated n times, and x 
equated to zero in the resulting expression. 

Substitute the above values of A, B, C, . . . , in (1) and we get 
dv x d' 2 v x 2 d B v x' 3 , c . 

"-^ai + j|n + ii + v (6) 

The series on the right-hand side is known as Maclaurin's Series. 
From (5), the series may be written, 

= /(O) + /(O)? + /"(O)^ + /"(0) 1 ^ 3 + . (7) 

* Note the resemblance between this expression ami (1) of the preceding section. 



228 HIGHER MATHEMATICS. 98. 

98. Useful Deductions from Maclaurin's Theorem. 

The following may be considered as a series of examples of the 
use of the formula obtained in the preceding section. Many of the 
results now to be established will be employed in our subsequent 
work. 

1. Tlie binomial tlworem. In order to expand any function by Maclaurin's 
theorem, the successive differential coefficients of u are to be computed and x 
then equated to zero. This fixes the values of the different constants. 

Let u = (a + x) n , 

dujdx = n(a + x) n - 1 , .'. / (0) = na n - J ; 

dtuldx* = n(n - 1) (a + x)"- 2 , .'. /"(O) = n(n - l)a n ~ 2 ; 

d 3 uldx* = n(n - 1) (n - 2) (a + a;)"- 3 , .-. /"(O) = n(n - 1) (n - 2)a"- :i , 
and so on. Now substitute these values in Maclaurin's series (6), 

(a + x) = a + 7 V - i* + "fo-^oH - *j* + . . . , . (1) 

a result known as the binomial series, true for positive, negative, or fractional. 
values of n. See page 22.* 

EXAMPLES. (1) Prove that 

(a - ar) = a - -a~ l x + n ^ n ~ 1 ^- 2 q- 2 - ... . (2) 

When n is a positive integer, and n = m, the infinite series is cut off at a 
point where n - m = 0. A finite number of terms remains. 
Establish the following results : 

(2) (1 + x 2 ) 1 ' 2 = 1 + x 2 /2 - a- 4 /8 + 6 /16 - ... 

(3) (1 - x 2 )- 1 / 2 = 1 + z 2 /2 + 3x 4 /8 + 5^/16 + . . . 

(4) (1 + x 2 ) ~ l = 1 - x 2 + x 4 - . . . 
Verify this last result by actual division. 

2. Trignometrical series. Suppose 

u = f(x) = sin x. 
Proceed as before. Note that 

<i(sin x)jdx = cos x t d(cos x)ldx = - sin x, etc. 
.-. sinO = 0, - sinO = 0, cosO = 1, - cosO = - 1. 

Hence, sin, . \ - * + * - g + . . . (3) 

A result known as the sine series. 

* In the proof that dx/dx = nx n ~ l , we have assumed the binomial theorem. 
The student may think we have worked in a vicious circle. This need not be. The 
result may be proved without this assumption. Let 

y = X M , X 1 = X + Sx, y j = y + Jy. 



by division. But Lt$ x = Q x l = x. 
' -3- 



98. INFINITE SERIES AND THEIR USES. 229 

' In the same way, show that 

cosx = l- *L + -!* + ...... (4) 

This is the cosine series. 

These series are employed for calculating the numerical values of angles 
between and |w. All the other angles found in " trignometrical tables of 
sines and cosines," can be then determined by means of the formulae, 

sin(r - x) = cos a? ; cos(ir - x) = sinx, 
of page 499. For numerical examples, see page 497. 

Now let u = f(x) = tan x. 

From page 499, .. u cos x = sin x. 

By successive differentiation of this expression, remembering that w x = duldx, 
21., = d^/fZ.r 2 , . . . , as in 8, 

.. z^cosx - usiux = coax. 

.*. w^cosx - 2zt 1 sinx - ucosx = - sinx. 

.. u 3 cosx - &i z siux - 3ttjCosx + usiux = - cosx. 

By analogy with the coefficients of the binomial development (1), or Leibnitz' 
theorem, 20, 

?* n cosx - -Un-fimx - -5 '-"- ^M n _ 2 cosx + . . . = nth derivative sinx. 

Now find the values of u, u lt u zt u s , ... by equating x = in the above 
equations, thus, 

/(O) =/"(<>)= ... =0 



Substitute these values in Maclaurin's series (7), preceding section. The 
result is, 



The tangent series. 

3. Inverse trignometrical series. Let 

6 = tan - J x. 
By (3), 15 and example (4) above, 

.-. dejdx = 1/(1 + x 2 ) = 1 - x 2 + x 4 - x 6 + . . . 
By successive differentiation and substitution in the usual way, 

tan-ix^x-^ + l- . . . , . .' . (6) 

or, from the original equation, 

6 = tan 6 - i tan :? + i tan 5 -...,. . . (7) 
which is known as Gregory's series. This series is known to be converging 
when 6 lies between - $v and J?r. 

Gregory's series has been employed to calculate the numerical valite of IT. 

Let e = 45 = |ir, . . x = 1. 

Substitute in (6), 

*= 1 - i i _ 1 !_.! .1 _ 
4 3 + 5 7 + 9 11 + 13 

The so-called Leibnitz series. This is a convenient opportunity to emphasize 
the remarks on the unpracticable nature of a slowly converging series. It 



230 HIGHER MATHEMATICS. 98. 

would be an extremely laborious operation to calculate v accurately by means 

of this series. A little artifice will simplify the method, thus, 

IT / ' 1\ /I 1\ /I 1 \ TT 2 2 2 



* - J_ J_ 1 

8 ~ 173 + 5.7 + 9.11 + *''' 

which does not involve quite so much labour. Itwill be observed that the angle x 
is not to be referred to the degree-minute-second system of units, but to the 
unit of the circular system (page 494), namely, the radian. Suppose x = I/ s 3, 
then tan -I a; = 30 = ?r. Substitute this value of x in (6), collect the positive 
and negative terms in separate brackets, thus 

_L i \ / i i 

/~ i c /T: ** "i" ' 



6 

To further illustrate, we shall compute the numerical value of IT correct 

to five decimal places. At the outset, it will be obvious that (1) we must 

include two or three more decimals in each term than is required in the final 

result, and (2) we must evaluate term after term until the subsequent terms 

can no longer influence the numerical value of the desired result. Hence : 

Terms enclosed in the first brackets. Terms enclosed in the second brackets. 

0-57735 03 0-06415 01 

0-01283 00 0-00305 48 

0-00079 20 0-00021 60 

0-00006 09 0-00001 76 

0-00000 52 0-00000 15 

0-00000 05 0-00000 02 



0-59103 89 0-06744 02 

.-. TT = 6(0-59103 -89 - 0-06744 02) = 3-14159 22. 

The number of unreliable figures at the end obviously depends on the 
rapidity of the convergence of the series (page 224). Here the last two figures 
are untrustworthy. But notice how the positive errors are, in part, balanced 
by the negative errors. The correct value of ir to seven decimal places is 
3-1415926. There are several shorter ways of evaluating TT. See Encyclopaedia 
Britannica, art. " Squaring the Circle ". 
We can obtain the inverse sine series 



. 

in a similar manner. Now write x = , sin - l x = ITT. Substitute these values 
in (8). The resulting series was used by Newton for the computation of TT. 
4. Exponential theorem. Show that 

v 7 ,2 r<& 

1+ I + Ii + fi + ........ < 9 > 

by Maclaurin's series. 

The exponential series expresses the development of e x , a x , or some other 
exponential function in a series of ascending powers of x and coefficients 
independent of x. 

EXAMPLES. (1) Show that if k = log a 

a ..i + te + w + w + ...... (io, 



INKIMTK SERIES AND THEIR USK- j:ll 



(3) Calculate the numerical value of e correct to four decimal places. 
Hint, put x = 1 in (9), etc. 

The development by Maclaurin's series cannot be used if the function 
or any of its derivatives becomes infinite or discontinuous when x is equated 
to zero. For example, the first differential coefficient of f(x) = \ f x, is $/ \'./:, 
which is infinite for x = 0, in other words, the series is no longer convergent. 
The same thing will be found with the functions log a:, cotx, 1/x, a 1 '* and 
sec ~ l x. Some of these functions may, however, be developed as a fractional 
or some other simple function of x, or we may use Taylor's theorem. 

99. Taylor's Theorem. 

Taylor's theorem determines tlie law for the expansion of a 
function of the sum, or difference of TWO variables * into a series 
of ascending powers of one of the variables. 

Now let 

Ui =/(# + y)> 
Assume that 

u, = f(x + y) = A + By + Gif + Df + . . . , . (1) 
where A, B, C, D, . . . are constants, independent of y, but 
dependent upon x and also upon the constants entering into the 
original equation. 

Differentiate (1) on the supposition that x is constant and y 
variable. Thus, 

fg = B + 2Cy + 9JW + . . . . (2) 

Now differentiate (1) on the supposition that y is constant and x 
variable, 

du, dA dB dC , dD 

~ = 2 - 



First, to show that 

du^ du 
<l>/ dx 
where u^ = f(x + y). 



* A function of the sum of two variables is such that if a single variable be 
substituted for that sum, the original function reduces to that of a single variable. 
For instance, 

sin x = u = sin (y + z), 
where ./: is the sum of the two variables y and z. 

f Note that du^dy and du^dx of (2) and (3) are really partial differential co- 
efficients. Strictly, we should write, 



232 HIGHER MATHEMATICS. 99. 

Now let 

v = x + y, .\ H! = f(v). 

Differentiate with respect to x, y constant ; also with respect to 
y, x constant. 

du^ du^ dv . du-^ du^ dv 
dx dv dx' dy dv dy 

(See page 29.) But v = x + y and if y is constant, dv = dx and 
dv/dx = 1 ; similarly, if x is constant, dv = dy, or dv/dy = 1, there- 
fore 

du\ _ du^ m du^ _ du^ di^ _ du^ 

dx dv ' dy ~ dv ' ' dx ~ dy ' 
It, therefore, follows that (2) and (3) are identical. 



Since this identity is true whatever be the value of y, the co- 
efficients of like powers of y, on each side of the equation, are 
equal each to each (footnote, page 172), therefore, 
dA dB dC 

c--*''H-* J ''"s- u >-> ..... ^ 

But, by hypothesis, (1) is true whatever be the value of y. We 
may, therefore, put y = so that the original equation reduces to 
a function of x, say, 

=/(*) ...... (6) 

A - u . B -^h. r 1*B' 1 *V n l d 1 **i- 
~dx' ( = 2S = 2'Sf' - D = 3'S = ^3-&3-' 

Substitute these values of A, B, C, D in the original equation 
and we obtain 

du y d 2 u y 2 d 3 u y* 
Ul = f(x + y} = u + Tx l + ^ JL_ + _ _!_ + . . . (7) 

The series on the right-hand side is known as Taylor's series. 
From (6), we may write Taylor's series in the form, 

, =f(x + y) =f(x)+f(x)l + f"(x)f- z +/' r y73 + (8) 
Or, interchanging the variables, 

! = /(* + y) - f(y) 



EXAMPLE. Prove that 

f(x - y) = f(x) - f'( 



Maclaurin's and Taylor's series are slightly different expressions 
for the same thing. The one form can be converted into the other 



$ '.<. INFINITE SERIES AND THEIR USES. -j:i:s 

In substituting f(x + y) for f(x) in Maclaurin's theorem, or by 
putting y = in Taylor's. The geometrical signification is that 
each function is the equation of a curve with a different origin 
on the o>axis and y denotes a constant, not an ordinate, on the 
abscissa axis. 

EXAMPLES. (1) Expand ?t, = (x + y) n by Taylor's theorem. Put y = 

and u = x", 

du cPu 

dx= 7l * n -' ; ^ = n < n - 1 >* n - 2 ' etc - 
Substitute the values of these derivatives in (8). 

.-. MJ = (x + y) n = x n + nx n ~ l y 
Verify the following results : 

(2) If k = log a, 

ttj = a* + y = a x (l + ky 

(3) Ul = (x + y + a) 1 ' 2 = (x + a) 1 ' 2 + \y(x + a) - 1 - . . . If x = - a, 
the development fails. 

(4) Wl = sin (x + y) = s'mx^l - |y + -|y - . . .) + cosx(y - |_ + . . .^. 
For numerical examples, see page 497. 

(5) log (x + y) = log x + ! - | r2 + jj. - . . . 

(6) log a (l +x) = to^(* - ^' 2 + |i" - 

(7) log (1 + y) = y - iz/ 2 + &f' - i?/ 4 + . . . 

(8) log (1 - y) =- (y + %y* + \if + iz/ 4 + ) 

If y = 1, the development gives a divergent series and the theorem is then 
said to fail. The last four examples are logarithmic series. 

(9) Put y = - x in Taylor's series, and show that 

x - :r 2 
/(*)=/(0) 



known as Bernoulli's series (of historical interest, published 1694). 

Mathematical textbooks, at this stage, proceed to discuss the conditions 
under which the sum of the individual terms of Taylor's series is really equal 
to f(x + y). When the given function f(x + y) is finite, the sum of the cor- 
responding series must also be finite, and the developed series (Taylor's or 
Maclaurin's) must either be finite or convergent. The development is said to 
fail when the series is divergent. 

It is not here intended to show how mathematicians have succeeded in 
placing Taylor's series on a satisfactory basis. That subject belongs to the 
realms of pure mathematics.* The reader must exercise " belief based on 
suitable evidence outside personal experience," otherwise known as faith. 
This will require no great mental effort on the part of the student of the 
physical sciences. He has to apply the very highest orders of faith to the 
fundamental principles the inscrutables of these sciences, namely, to the 

* If the studeiit is at all curious, Todhunter, or Williamson on " Lagrange's 
Theorem on the Limits of Taylor's Series," is always available. 



234 HIGHER MATHEMATICS. * W. 

theory of atoms, stereochemistry, affinity, the existence and properties of 
interstellar ether, the origin of energy, etc., etc. What is more, " reliance on 
the dicta and data of investigators whose very names may be unknown, lies 
at the very foundation of physical science, and without this faith in authority 
the structure would fall to the ground ; not the blind faith in authority of 
the unreasoning kind that prevailed in the Middle Ages, but a rational belief in 
the concurrent testimony of individuals who have recorded the results of their 
experiments and observations, and whose statements can be verified . . .".* 

The theory of proportional parts or proportional differences is 
an application of Taylor's theorem. If a small number be in- 
creased by a small fraction of itself, the increase in the value of 
the number is nearly proportional to the increase of its logarithm. ) 
Thus, 

Iog 10 (w + h) = Iog 10 w(l + -J = Iog 10 w + Iog 10 (l + 



For example, let n be not less than 10,000 and h not greater 
than unity, h/n is not greater than O'OOOl and the next term is not 
greater than i(O'OOOl) 2 , that is to say, not exceeding 0*000000,0025. 
The next term is, of course, much less than this. We may, there- 
fore, correctly write, as far as seven decimal places, 

log (n + h) - log n = 0-4343 x h/n 
and log (n + 1) - log n = 0-4343 x l/n. 

By division, we get the important result, 

log (n + h) - log n = h 

log(n + 1) - logw 1' ' 

provided the differences between two numbers n and h are such 
that n is of the order of 10,000 when x is less than unity. 

This formula, known as the rule of proportional parts, is 
used for finding the exact logarithm of a number containing more 
digits than the table of logarithms allows for, or for finding the 
number corresponding to a logarithm not exactly coinciding with 
those in the tables. The following examples will make this clear : 

* Excerpt from the Presidential Address of Dr. Carrington Bolton to the Washing- 
ton Chemical Society, English Mechanic, 5th April, 1901. 

f This is commonly stated as an exercise on differentiation. A question like this. 
is set : " How much more rapidly does the number x increase than its logarithm { " 
Here d(log x)/dx = l/x. The number, therefore, increases more rapidly or more slowly 
than its logarithm according as x > or < 1. If x = 1, the rates are the same. If 
common logarithms are employed, M ( 16) will have to be substituted in place of 
unity. E.y., d(\og lo x)dx = M/x. 



INFINITE SKRIKS AND TIIKIK I'SKS. 



236 






EXAMPLES. (1) Find the logarithm of 46502-32, having given 
log 46501 = 4-6674623 
log 46502 = 4-6674716 

Difference = 0-0000093 

Let x denote the quantity to be added to the smaller of the given logs. 
The problem may be stated thus, 

log n = log 46501 = 4-6674623 ; 

log (n + 1) = log (46501 + 1) = 4-6674623 + 0-0000093 ; 
log (n + h) = log (46501 + x) = 4-6674623 + jc, 

by simple rule of three : if a difference of 1 unit in a number corresponds with 
a difference of 0-0000093 in the logarithm, what difference in the logarithm 
will arise when the number is augmented by 0*32 ? 

.-. 1 : 0-32 = 0-0000093 : x, .-. x = 0-00000298 . . . 
The required logarithm is, therefore, 4-6674653. 

(2) Find the number whose logarithm is 4-6816223, having given 

log 48042 = 4-6816211 ; log 48043 = 4-6816301. 

Since a difference of unity in the number causes a difference of 0-0000090 
in the logarithm, what will be the difference in the number when the logarithms 
differ by 0-0000012 ? 

.-. 1 : x = 0-0000090 : 0-0000012, 

x = 0-13, or the number is 48042-13. 

The rest of this chapter will be mainly concerned with direct 
or indirect applications of infinite converging series. 183 on 
proportional errors and 158 on the use of Taylor's theorem in 
finding the approximate roots of an equation, may also be consulted. 

100. The Contact of Curves. 

The following is a geometrical illustration of one meaning of the different 
terms in Taylor's development. 

If four curves Pa, Pb, PC, Pd, . . . (Fig. 101), have a common point P, 
any curve, say PC, which passes between two others, Pb, Pd, is said to have a 
closer contact with Pb than Pd has. 



/Y 




FIGS. 101, 102. Orders of Contact of Curves. 



Now let two curves P^ and P^ (Fig. 102) referred to the same rect- 
angular axes, have equations, 

and, y, =/!(*,). 



236 HIGHER MATHEMATICS. S 101. 

Let the abscissa of each curve at any given point, be increased by a small 
amount 7t, then, by Taylor's theorem, 



If the curves have a common point P , x = x l and y = y l &t the point of 
contact. Substitute the coordinates of this point in equations (1) and (2). 
f(x + h) will represent the ordinate PMj and fi(x l + h), the ordinate P^M r 
Similarly, dyjdx, d^y/dx 2 . , . will represent the differential coefficients of the 
ordinate of the curve f(x + h) at the point P ; dyjdx^ d^y^dx\ . . . , similar 
properties for the second curve fi(x l + h). 

Since the first differential coefficient represents the angle made by a 
tangent with the ic-axis, if, at the point P , 

x = x l ; y = y l and dy/dx = dyjdx^ 

the curves will have a common tangent at P . This is called a contact of the 
first order. If, however, 

x = x lt y = y l ; dyjdx = dy l jdx l and cPyjdx* = d?y^\dx\, 

the curves are said to have a contact of the second order, and so on for the 
higher orders of contact. 

If all the terms in the two equations are equal the two curves will be 
identical ; the greater the number of equal terms in the two series, the closer 
will be the order of contact of the two curves. 



101. Extension of Taylor's Theorem. 

Taylor's theorem may be extended to include the expansion of 
functions of two or more independent variables. Let 

u=f(x,y\ . . . (1) 

where x and y are independent of each other. Suppose each 
variable to change independently so that x becomes x + h and 
y becomes y + k. 

Let f(x, y) change to f(x + h, y). By Taylor's theorem 



If y now becomes y + k, each term of equation (2) will change so 
that 

~du-, l^ 2 u k' 2 

u becomes u + k + i _+...; 
ty l)y 2 2! 

*bu , Du Wu 7 <)% . Wu Wu 7 

becomes + -k +...;_ becomes - + - k + . . . , 



by Taylor's theorem. Now substitute these values in (2) and we 
obtain, 



INFINITE SERIES AND THEIR USES. J:-7 



^u, d 2 w h 

+ 5? + s* ar + ) 

Su=f(x + h, y + k) -f(x,y); 

. . . (3) 



The final result is exactly the same whether we expand first 
with respect to y or in the reverse order. 

By equating the coefficients of hk in the identical results ob- 
tained by first expanding with regard to h, (2) above, and by first 
expanding with regard to k, we get 



which was obtained another way in 23. 

The investigation may be extended to functions of any number 
of variables. 

EXAMPLE. Show that 



102. The Determination of Maximum and Minimum Values 
of a Function by means of Taylor's Series. 

I. Functions of one variable. Taylor's theorem is sometimes 
useful in seeking the maximum and the minimum values of a 
function, say, y = f(x). It is required to find particular values of 
x in order that y may be a maximum or a minimum. 

Let x change by a small amount h so that by Taylor's 
development, 



First, it must be proved that h can be made so small that C -^h will be 

greater than the sum of all succeeding terms of Taylor's series. Assume 
that Taylor's series may be written, 

f(x + h} = y + Ah+ Bh? + CW + . . . , 

where A, B, C, . . . are coefficients independent of h but dependent upon x, 
then, if Rh = Bh + CW + ... we have, 

f(x + h) =y + h(A + Eh) ..... (2) 

For sufficiently small values of h, Rh must be less than J, because, by 
hypothesis, .4 is independent of h. 



238 HIGHER MATHEMATICS. 102. 

Second, when x changes by a small amount h, it follows, ex- 
amples 57, that for a maximum, f(x) >f(x + h), and/(#) >f(x - h) ; 
for a minimum, f(x) < f(x + h), and f(x) < f(x - h). It is, therefore, 
easy to see that if 

fis negative, f(x) will be a maximum ; 
f(x h) - /(#h is positive, f(x) will be a minimum ; 

Ichanges sign, /(a?) will be neither. 
whatever the sign of ft.* 

If dyldx has a finite value, h may be imagined so small that 
the sign of A + Eh of (2) does not change when that of ft changes. 
Therefore, the sign of f(x + ft) - f(x) will depend on that of ft, 
and consequently f(x) cannot be either a maximum or a minimum. 
Only when ft and A + Eh change sign simultaneously (as ft passes 
through zero) can x be either a maximum or a minimum. Under 
these circumstances, dyldx becomes zero for maximum or minimum 
values of y. 

If dx/dy vanishes, 



As before, it can be shown that -JjJ is greater than all suc- 

ceeding terms of the series. But ft is of necessity positive, the 
sign of the second differential coefficient will, therefore, be the 
same as that of f(x + ft) - f(x). In other words, y will be a 
maximum when dyldx = and d^yldx^ is negative, and a minimum, 
if d^yldx^ is positive. 

If, however, the second differential coefficient vanishes, the 
reasoning used in connection with the first differential must be 
applied to the third differential coefficient. If the third derivative 
vanishes, a similar relation holds between the second and fourth 
differential coefficients. 

To generalise, if the order of the first differential coefficient that 
does not vanish is odd, f(x) will be neither a maximum nor a 
minimum unless d n y/dx n passes through infinity (where n is the 
order of the differential that does not vanish). If n is even, we 
shall have a maximum or a minimum according as d n yldx" is 
negative or positive. 

* When reference is made to a magnitude without reference to its positive or 
negative values it is frequently written \h\, \a\, |sina;|, and called the absolute rvalue 
of h, a, or sin a;, as the case might be. In this work h is written for \h\, + sinx 
for I sin x j, etc. 



< lo INFINITE SMK IKS AND TIIKIK USES. !>:','. 

Hence the rules : 

1. y is either a maximum or a minimum for a given value of x 
only when the first non-vanishing derivative, for this value of x t is 

000ft, 

2. y is a maximum or a minimum according as the sign of the 
non-vanishing derivative of an even order, is negative or positive. 

In practice, if the first derivative vanishes, it is often con- 
venient to test by substitution whether y changes from a positive 
to a negative value. If there is no change of sign, there is neither 
a maximum nor a minimum. 

EXAMPLES. (1) Test y = x?- 12a; 2 -60.r for maximum or minimum values. 
dyldx = 3x 2 - 24x - GO ; .-. x* - Sx - 20 = 0, or x = - 2, or + 10. 
<Pyjdx z = 6x - 24 ; or, x = + 4. 

Since d 2 y/dx 2 is positive when x = 10 is substituted, x = 10 will make y a 
minimum. When - 2 is substituted, d 2 y/dx 2 becomes negative, hence x = - 2 
will make y a maximum. This can easily be verified : 

If x = - 3, - 2, - 1, . . . + 9, +10, + 11, ... 

y = + 45, +64, +48, ... - 783, - 800, - 781 ... 

(max.) (min.) 

(2) What value of x will make y a maximum or a minimum in the ex- 
pression, y = x' 3 - 6x 2 + llx - 6 ? 

dyldx = 3z 2 - I2x + 11 = ; .-. x = 2 I/ s/3 ; 
(Py/dx* = 6* - 12. 

If x = 2 + I/ Vs", d*yldx 2 = 2 \/3. . (max.) ; 

x = 2 - I/ \/3, dPy/dx* = - 2 ^3. . (min.). 

II. Functions of tioo variables. To find particular values of 
x and y which will make the function, 

u = f(x, y), 

a maximum or a minimum. As before, when x changes by a small 
amount h, and y by a small amount k, if 

/is negative, f(x, y) will be a maximum ; 
f(x h, y k) - f(x, y)) is positive, f(x, y) will be a minimum ; 

[changes sign, f(x, y) will be neither, 
whatever be the signs of h and k. Also, let 

u = f(x + h,y + k) - f(x, y). 

Expand this function as in the preceding section (3). By making 
the values of h and k small enough, the higher orders of differ- 
entials become vanishingly small. But as long as ~bu/?)x and 
remain finite, the algebraic sign of &u will be that of 



240 HIGHER MATHEMATICS. 102. 

At a maximum or a minimum point, we must have 



and, since h and k are independent of each other, u can have a 
maximum or a minimum value only when 

l)u ~du 

5 _ = and 3 -=0 ..... (5) 

because the sign of Su, in (4), depends upon the signs of h and k. 
Thus Su will be positive for some values of Duftx, negative for 
others. The same thing holds for 'bufty. Substituting ^uj^x = 0, 
= in (3), 101, we get 



If h and k be taken sufficiently small, 8u will always have the 
same sign. (Why?) For the sake of brevity, write the homo- 
geneous quadratic (6) in the form 

ah 2 + bhk + cfc 2 . 

On page 388, it is shown that the sign of this quadratic remains 
invariable, provided ac is greater than 6 2 and the signs of a and 
c are the same. This means that if condition (5) holds, &u will 
have the same sign for all values of h and k within certain limits, 
provided ^ufox 2 and Wufby' 2 have the same sign and 

Wu Vu . - 

^T, ^7 is greater than 
2 2 



This is Lagrange's criterion for the maximum and minimum 
values of a function of two variables. When this criterion is 
satisfied, f(x, y) will be a maximum or a minimum according as the 
sign of ^' 2 ul^x 2 (or ^ 2 u/^y' 2 ) is negative or positive. 

d% Wu . 
If 



or WulDx 2 and Wufty' 2 have different signs, the function is neither a 
maximum nor a minimum. There is a point of inflection. 

**u >% Vu_ 
2 * ' 



in order that a maximum or a minimum may occur, it is neces- 
sary that the first set of differential coefficients which do not 
vanish, shall be of an even order. 



102. 



INFINITE SERIES AND THEIR USES. 



241 



EXAMPLES. (1) Test the function u = r 1 + y 3 - Baxy for maxima or 
minima, 

'duj'dx = 8z 2 - Say = 0, .-. y = x 2 /a ; 

'du/'dy = By 2 - Bax = 0, .-. y 2 - ax=ar/a 2 - ax = 0. 

.. x = 0, x 3 - ft 3 = 0, or x = a. 
The other roots, being imaginary, are neglected. 

.. y = 2 /ft = ft, or T/ = 0. 
3% 3% B 2 u 



Call these derivatives (ft), (6), and (c) respectively, then 
If x = 0, (ft) = 0, (6) = - 3ft, (c) = 0. 

If x = ft, (ft) = 6ft, (b) = - 3ft, (c) = 6ft. 

^-^:. = - 3ft. 



which means that x = a will make the function a minimum because 
is positive ; x = will give neither a maximum nor a minimum. 

(2) Find the condition that the rectangular parallelepiped whose edges 
are x, y and z shall have a minimum surface when its volume is v 3 . 

Since v 3 = xyz, u = xy + yz + zx = xy + v 3 /x + v 3 /y. When 9w/3x = O r 
x'*y = v 3 ; when 'du/'dy 0, xy 2 = v 3 . The only real roots of these equations- 
are x = y = v, therefore, z = v. The sides of the box are, therefore, equal to 
each other. 

(3) Show that u = a*y*(l - x - y) is a maximum when x = \, y = \. 

(4) Find the maximum value of u in u = x 5 - Soar 2 - 4ft?/ 2 . 'du/'dx = Bx(x - 2a) ; 



= - 8ft. Condition (5) 



=-8ay; 'Puffx* = 6(x - a) ; o 2 u/'dx'dy = ; 
is satisfied by x = 0, y = and by x = ft, y = 0. 
The former alone satisfies Lagrange's condition 
(7), the latter comes under (8). 

(5) In Fig. 103, let P l be a luminous point ; 
OM it OM<i are mirrors at right angles to each 
other. The image of P 1 is reflected at jVj and 
JV.J in such a way that (i.) the angles of inci- 
dence and reflection are equal, (ii.) the length 
of the path P^N^ is the shortest possible. 
(Fermat's principle, " a ray of light passes from 
one point to another by the path which makes 
the time of transit a minimum ".) Let ^ = r lt 
i 2 = r 2 be the angles of incidence and reflection 
as shown in the figure. To find the position of 

Let CW = x; ON l = y, OM 2 = c^ ; M 




and N z . 

= 6 2 ; OM l 




Let 



.'. S = 

Find 9s/dz and dscty. Equate to zero, etc. Ansr. 

x = (a^ - ft 1 6 2 )/(6 1 + 62) ; y = (^ - 1 6 2 )/(ft 1 + 0,2). 

Note that x/y = (ftj + fta)/(6i + & 2 ). Work out the same problem when the 
angle M^OM^ = a. 

(6) Required the volume of the greatest rectangular box that can be sent 

Q 



242 HIGHER MATHEMATICS. $ 103. 

by " Parcels Post " in accord with the regulation : " length plus girth must 
not exceed six feet". Ansr. 1ft. x 1 ft. x 2ft. = 2c.ft. Hint. V = xyz is to 
be a maximum when V = x + 2(y + z) = 6. But obviously y = z, .-. V = xy z 
is to be a maximum, etc. 

(7) Required the greatest cylindrical case that can be sent under the same 
regulation. Ansr. Length 2ft., diameter 4/ft., capacity 2*55 c.ft. Hint. 
Volume of cylinder = area of base x height, or, ^irW 2 is to be a maximum 
when the length + the perimeter of the cylinder = 6, i.e., I + irD = 6. Ob- 
viously I and D denote the respective length and diameter of the cylinder. 

See also S 106. 



103. Indeterminate Functions.* 

In discussing the velocity of reactions of the second order, we 
found that if the concentration of the two species of reacting 
molecules is the same, the expression 

ftt = ^_log?JL.( 

a - b b - x b' 

assumes the indeterminate form 

kt = CD x 0, 

by substituting a = b. W T e are constantly meeting with the same 
sort of thing when dealing with other functions, which may reduce 
to one or other of the forms : 0/0, GO/ oo, oo - GO, I 00 , 00, ... 
We can say nothing at all about the value of any one of these 
expressions, and, consequently, we must be prepared to deal with 
them another way so that they may represent something instead 
of nothing. They have been termed illusory, indeterminate and 
singular forms. 

Fractions which assume the form J are called vanishing 
fractions, thus, (ax" - %a 2 x + a s )/(bx' 2 - 2abx + bd 2 ) reduces to g, 
when x = a. The trouble is due to the fact that the numerator 
and denominator contain the common factor (x - a) 2 . If this is 
eliminated before the substitution, the true value of the fraction 
for x = a can be obtained, viz., a/b. 

These indeterminate functions may often be evaluated by alge- 
braic or trignometrical methods, but not always. Taylor's theorem 
furnishes a convenient means of dealing with many of these func- 
tions. The most important case for discussion is "," since this 

* In one sense, the word "indeterminate " is a misnomer, because it is the object 
of this section to show how values of such functions may be determined. 



s lo:j. INFINITE SERIES AND THEIR USES. 243 

form most frequently occurs and most of the other forms can be 
referred to it by some special artifice. 

Case i. The function assumes the form g. As already pointed 
out, the numerator and denominator here contain some common 
factor which vanishes for some particular value of x, say. These 
factors must be got rid of. One of the best ways of doing this, 
short of factorising at sight, is to substitute a + h for x in the 
numerator and denominator of the fraction and then reduce the 
fraction to its simplest form. In this way, some power of h will 
appear as a common factor of each. After reducing the fraction 
to its simplest form, put h = 0, so that a = x. The true value of 
the fraction for this particular value of the variable x will then 
be apparent. 

For cases in which x is to be made equal to zero, the numerator 
and denominator may be expanded at once by Maclaurin's theorem 
without any preliminary substitution for x. For instance, the trig- 
nometrical function (sin x)/x approaches unity when x converges 
towards zero. This is seen directly. Develop sin a? in ascending 
powers of x as indicated on page 228. We thus obtain 

(/> o*3 /y5 /y7 \ 
? _ _ + _ + \ 
1 3! 5! 7! ' ' ') = -, _ x*_ x_ _ x* 

x x 3l + 5l ~ 7l 

The terms to the right of unity all vanish when x = 0, therefore, 






X 

EXAMPLES. (1) Show Lt x = (a x - 6*)/;r = log a/6, page 37. 

(2) Show Mr = (l ~ cosz)/z 2 = . 

(3) The fraction (x" - a n )/(x - a) becomes 0/0 when x = a. Put x = a + h 
and expand by Taylor's theorem in the usual way. Thus, 

T . x n ci n T . (a + h) n a n 

Lt x = a = fo = a - { = na"- 1 , etc. 

x - a h 

It is rarely necessary to expand more than two or three of the lowest powers 
oih. 

(4) Show Lt x = !*-^-^ = -. Put x = 1 + h and expand. 

J. X It 



* The symbol " x = " is sometimes used for the phrase " as x approaches zero ". 
_ Q or ._ Q are also used instead of our "Zr = " meaning "the limit of 
... as x ^approaches zero ". 



244 HIGHER MATHEMATICS. $ 103. 

The following method will be found very convenient for dealing 
with indeterminate functions of this nature. 

Let ^y^y - be a fraction in which /(a?), f^x) are the parts 

fi( x ) ( x ~ a ) 

not containing vanishing factors, n and m are positive integers. 
First let m = n. By differentiation of the numerator and deno- 
minator m times and then substituting a for x we get /(xj/'f^x). 
For instance, the value of the expression, 

Lt X '~ 1 

* =1 



is obtained by differentiating once so that 

Lt x = * 

= 1 Qx 2 + x - 2 ~ 5* 

If n > m, the numerator will vanish after m differentiations 
and the fraction will be equal to zero. If n < m, the denominator 
will vanish after n differentiations and the fraction will become 
infinitely great. Under these circumstances we proceed by the 
following rule : To find the value of an algebraic fraction, sub- 
stitute the successive differential coefficients of the numerator and 
denominator until a numerator and denominator are obtained 
which do not vanish for the value of x under consideration. If 
the numerator vanishes when the denominator is finite, the 
limiting value is zero. Some transcendental functions may be 
treated in this way. 

EXAMPLES. (1) Prove that / = logic, by means of the general formula 

\x n dx = flLLi. Hint. Show that 
n + l 

/y.W + 1 

Ltn^-i- - = log or. 

n + 1 

Differentiate the numerator and denominator separately with regard to n and 
substitute n = - 1 in the result. See 71. 

(2) Show Lty = 1 (* - 1 ) = c log l \ See (4), 91, (8), 92. 
7 - J-V^ 7 v r I v i 

Case ii. The function assumes the form oo/ oo. These can be 
converted into the preceding "J" case by interchanging the 
numerator and denominator, or else proceed as for g. by the 
method of successive differentiation. 

EXAMPLES. (1) Show that 

log sin x 



103. I N FINITE SERIES AND THEIR USES. i>45 

Simple substitution furnishes - oo/- oo. Differentiate once and 

cos a- sin2x _ 1 
2 cos 2x s'mx ~ 2 * 0' 
Differentiate the second factor once more and we get 

2 cos 2a-/cos x = 2, etc. 
(2) Lt x -e x lx^ = 00/00, when n is positive. Differentiate n times and 



1.2 ... n "-* 

Case iii. T/ie function assumes the form oo x 0. Obviously, 
such a fraction can be converted into the "0/0" form by putting 
the infinite expression as the denominator of a fraction. 

EXAMPLES. (1) x log x becomes x - oo, when x = 0. Transpose the 
infinite term to the denominator and differentiate. 

or/logo: becomes on differentiation z 2 ; .'. Lt x = x\ogx = 0. 

(2) Show Lt a = b _L_ log j* - *f = _^_ 

a - b (b - x)a a(a - x) 

(3) Show Lt x = e~ x \og x = x oo = 0. 

Case iv. The function assumes the form GO oo. First reduce 
the expression to a single fraction and treat as above. 

EXAMPLES.-(I) Lt.^r _ J_ = slogs-s-1. 

l x - 1 log x (x - 1) log x 

Differentiate twice and 

T , x 1 

Lt ' ='*T3 -5- etc ' 



Case v. T/ie function assumes one of the forms 1, 00, 0. 
Take logarithms and proceed as above. 

EXAMPLES. (1) Lt x = x x = 0. Take logs and the expression becomes 
- oo/ oo ; differentiate and Lt x = x x = 1. 
(2) Show Lt x = Q (l + mx} llx = I 00 = c m . 

Sometimes a simple substitution will make the value apparent 
at a glance. For instance, substitute x = \ly and show that 
T . x + a T . I + ay -, 

Jjt.-a, - j- = L/t u - n - -- * = I- 

-~x + b -1 + by 

Another illustration has been studied in 16, namely, 



246 HIGHER MATHEMATICS. $ 104. 

103. " The Calculus of Finite Differences." 

In the series, 

I 3 , 23, 3 3 , 4 3 , 5 3 , . . . , 

subtract the first term from the second, the second from the third, 
the third from the fourth, and so on. The result is a new series, 

7, 19, 37, 61, 91, . . . , 

called the first order of differences. By treating this new series 
in a similar way, we get a third series, 

12, 18, 24, 30, . . . , 

called the second order of differences. This may be repeated as 
long as we please, unless the series terminates or the differences 
become very-irregular. 

The different orders of differences are usually arranged in the 
form of a " table of differences ". To construct such a table, it is 
usual to begin at some convenient place towards the middle of a 
series of corresponding values of the two variables, to denote the 
different values of one variable by, say, 

and corresponding values of the other by, say, 

ni ni ni nt ni 

The differences between the independent variables are denoted by 
the symbol "A," with a superscript to denote the order of differ- 
ence and a subscript to show the relation between it and the 
independent variable. Thus, 

Argument. Function. Orders of Differences. 

x -fr 2/-2> Al 

r <n ~ 2 ' A* 

*-li i/-l> Al T -2' A3 

*o 2/o A2 -i> A4 -2> 

T ' 7/' A ' A* A ~ P 

xl yl ' AV 

where 

' A 2 = 2/2 ~ %i + 2/0' e ^ c> 

Such a table will often furnish a good idea of any sudden change 
in the relative values of the variables with a view to expressing the 
experimental results in terms of an empirical or interpolation for- 
mula. It is not uncommon to find faulty measurements, and other 
mistakes in observation or calculation, shown up in an unmistakable 
manner by the appearance of a marked irregularity in a member 
of one of the difference columns. It is, of course, quite possible 



104. INFINITE SERIES AND THEIR USES. 247 

that these irregularities are due to something of the nature of a 
discontinuity in the phenomenon under consideration. 

To find the differential coefficients of one variable with respect 
to another from a table of differences. If corresponding values of 
two variables can be represented in the form of a mathematical 
equation, the differential coefficient of the one variable with respect 
to the other can be easily obtained. If an empirical formula is 
not available, the tangent to the "smoothed" curve, obtained by 
plotting the corresponding values of x and y on coordinate or 
squared paper, will sometimes allow the differential coefficient to 
be deduced but not always. 

According to Stirling's interpolation formula, 

X ' 



y-y + - , 

' y ' ~~~ 



I' ~2~~ 2! * ~3! ~ 2 



(J. Stirling's Methodus Differentialis, London, 1730), when we 
are given a set of corresponding values of x and y, say # , y Q ; 
x v y v . . . , we can calculate the value y corresponding to any 
assigned value x, lying between # and x r (This kind of opera- 
tion is discussed in the next section.) Stirling's interpolation 
formula supposes that the intervals x l # , # - aj_ lf . . . are 
unity. If, however, h denotes the equal increments in the values 
x l - X Q , X Q - x_ l . . ., Stirling's formula is written 



x A> + Ai a* (x + h)x(x-h) 

~ 



+snp 4 - i+ si 



h)x*(x-h) 



A - 



(1) 



(x + 2fe) (x + h)x(x - h) (x - 2/t) A 5 _ 2 + A 5 _ 3 

5! k* 2, 

Differentiate (1) with respect to x. Put x = in the result 
<fy_V Al o + Al -i 1 A3. 1 + A3_ 2 1 A^_ 2 + A^ 3 _ \ 

dx~h\ 2 6" 2 + 30' 2 

This series may be written in the form, 
^//_1/A 1 + A 1 _ 1 P A^_ 1 + A 3 _ 2 1^.2' A 5 
dx~h\ 2~ 3!" 2 5! ' 

The following method of deducing (2) is instructive. Assume the expansion 

y = A + Bx + Cx z + Dy? + Ex 4 + . . . = say, T/ O . 
Differentiate with respect to x, 

.-. dyjdx = B + 2Cx + 3Dx* + lEx* + ..... (4) 



248 HIGHER MATHEMATICS. 104. 

Let x receive a small increment + h and also a small increment - h, then, 
from|the original equation, 

y = A + B(x + h) + C(x + li) z + D(x + h) 3 + . . . = say, y l ; 
y = A + B(x- h) + C(x - 7i) 2 + D(x - h)* + . . . = say, y_ t . 
But 2/i - 2/o = Al o; 2/o ~ 2/-i = A 1 -!, . . . 

Therefore, making the obvious subtractions, 

A 1 ,, = Bh + C(2xh + 7i 2 ) + D^xVi + 3z/i 2 + h 3 ) + . . . 
A 1 _ 1 = Bh + C(2xh - 7z 2 ) + D(9aflh - 3xW + h 3 ) + . . . 
Add these equations and divide by 27z, 



^ 

h 2 

When h is made very small, the terms containing h may be neglected. The 
resulting'series, 

Lt inCr ' y = Al + Al ~ 1 - = Lt k = { 
incr. x 2 h 

= B + 2C 



is identical with that just developed for dyjdx in equation (4). As a first 
approximation, therefore, we can write 

dy_l AJp + A 1 -! (5) 

dx ~ h ' 2 

If a greater accuracy than this is desired, substitute x + 2h and x - 2h for 
x in the original equation. In this way we can build up (2). 

To illustrate the use of formula (2), let the first two columns 
of the following table represent a set of measurements obtained 
in the laboratory. It is required to find the value of dy/dx 
corresponding to x = 5' 2. 

X y A 1 A 2 A 3 A 4 
4-7 109-947 n 563 
4-8 121-510 . 1-217 . 126 



4-9 134-290 14 . 123 1-343 . U3 0-017 

5-0 148-413 1-486 0-012 

5-1 164-022 1-641 0-019 

5-2 181-272 . 1-815 . 0-015 



5-3 200-337 23 . Q69 2-004 0-024 

5-4 221-406 2-217 Q . 231 0-018 

244 ' 692 25-734 2 ' 448 0-259 ' 028 
5-6 270-426 28>441 2-707 

5-7 298-867 

Make the proper substitutions in (2). In the case of 5*2 only the 
block figures in the above table are required. Thus, 
dy = 1 /17-250 + 19-Q65 1 0-174 + 0-189 1_ Q-QQ9 - Q-Q04\ 
dx 0-1V 2 6" 2 + 30" 2 

= 181-273. 

The second and third terms are not often used. They have the 
nature of correction terms. 



INFINITE SERIKS AND THKIK !-!> 

In the same way it can be shown by differentiating (1) twice, 
and putting x = 0, 

l A 4 , * A \ (* } 

- 1 ir- + r 7 



1/2. 2 2. 2.2-., 2. 2'. 3* 
- - '- "- 



EXAMPLES. (1) From Horstmann's observations on the dissociation pres- 
sure (/>) of the amuionio-chlorides of silver at different temperatures (0) : 
= 8, 12, 16, ... C. 

p = 43-2, 52-0, 65-3, . . . cm. Hg., 
show that at 12, dpjde = 2-76. 

(2) Find ds/de at C. from the following data : 

0=1, 0-5, 0, -0-5, -1-0, . . . ; 

10 6 x s = 1288-3, 1290-7, 1293-1, 1295-4, 1297-8, . . . 
Ansr. ds/de = 5-7 x 10 6 . 

(3) Find the value of cPyldx* for y = 5-2 from the above table. Ansr. 181-37. 
Also plot the dyjdx, 7/-curve from the data given. 

The difference columns should not be carried further than is consistent 
with the accuracy of the data, otherwise the higher approximations will be 
less accurate than the first. Do not carry the differences further than the 
point at which they begin to exhibit marked irregularities. 

(4) The variation in the pressure of saturated steam (p) with temperature 
(tf) has been found to be as follows : 

e = 90, 95, 100, 105, 110, 115, 120, . . . ; 

p = 1463, 1765, 2116, 2526, 2994, 3534, 4152, . . . 
Hence show that at 105 dpjde = 87-6, cPpldP = 2-48. 

Everett's papers in the Quarterly Journal of Pure and Applied Mathe- 
matics, 30, 357, 1900 ; 31, 304, 1901, may be consulted for some recent work 
on this subject. See also Nature, 60, 271, 365, 390, 1899. 



105. Interpolation and Empirical Formulae. 

After a set of measurements of two interdependent variables 
has been made in the laboratory, it is necessary to find if there is 
any simple relation between them, that is, to find if a general ex- 
pression of the one variable can be obtained in terms of the other 
so that intermediate values can be calculated. The process of 
computation of the numerical values of two variables intermediate 
between those actually determined by observation and measure- 
ment, is called interpolation. When we attempt to obtain values 
lying beyond the limits of those actually measured, the process is 
called extrapolation. 

It is apparent that the correct formula connecting the two vari- 



250 HIGHER MATHEMATICS. 105. 

ables must be known before exact interpolation can be performed, 
so much so that the method of testing a supposed formula is to 
compare the experimental values with those furnished by interpo- 
lation as exemplified in 18, 88, 96 and elsewhere. 

Interpolation is based on the fact that when a law is known 
with fair exactness, we can, by the principle of continuity, antici- 
pate the results of any future experiments. 

If only two experimental results are known, we must assume 
that the two quantities vary in a proportional manner. The 
geometrical meaning of this is that if the positions of two points 
are known, we must assume that the curve passing through these 
points is a straight line, because an infinite number of curved 
lines could be drawn through these two points. 

If the differences bet\veen the succeeding pairs of values are 
small and regular, any intermediate value can be calculated by 
simple proportion on the assumption that the change in the value 
of the function is proportional to that of the variable. Interpola- 
tion is employed in the graduation of a thermometer between 
and 100, extrapolation beyond these points. In Gauss' method of 
double weighing, the mean weight of the substance weighed in each 
pan is regarded as the true weight. 

The position of rest of a balance is deduced from the amplitude 
of the oscillations on each side. Three, five, or some odd number 
of observations are made, the arithmetical mean of the observations 
on each side are added together, the mean of this sum is the null 
point, or position of rest of the balance. 

Weighing by the method of vibrations is another example of 
interpolation. Let x denote the zero point of the balance, let w^ 
be the true weight of the body in question. This is measured by 
the weight required to bring the index of the balance to zero point. 
Let #! be the position of rest when a weight w l is added and x 2 
the position of rest when a weight w. 2 is added. Assuming that 
for small deflections of the beam the difference in the two positions 
of rest will be proportional to the difference of the weights, the 
weight (WQ) necessary to bring the pointer to zero will be given by 
the simple proportion : 

K - w i) ( x o ~ x i) = ( w -2 ~ w i) ' ( x -2 ~ x i)> 
or, w = w l + (x - xj (w 2 - w l )/(x 2 - xj. 

When the intervals between the two terms are large, or the 



106, INFINITE SERIES AND Til K IK I'SES. i'.M 

differences between the various members of the series decrease 
rapidly, this simple proportion cannot be used with confidence. 

To take away any arbitrary choice in the determination of the 
intermediate values, it is commonly assumed that the function can 
be expressed by a limited series of powers of one of the variables. 
Thus we have the interpolation formulae of Newton, Bessel, Stirling, 
Lagrange, and Gauss. 

Let x_. 2 , y_., ; x_ l} y_ l ; x , t/ ; x lt y 1 ; x 2 , y. 2 be corresponding 
values of the two variables x and y. It is required to calculate the 
value y corresponding to some value x lying between rr and x r 

Newton's interpolation formula is 



2/0 



continued until the differences become negligibly small or irregular. 

EXAMPLE. The use of Newton's formula may be illustrated by the follow- 
ing problem: What is the cube root of 60 '25, given the first two columns in 
the subjoined table ? The cube root of 

60 = 3-914868 , = . 021629 

61 . 3-936497 ^ I . 021394 A - - 0-000235. 

62 = 3-957891 > " J ^ = - 0-000228. 

63 = 3-979057 = . 020943 A'^ 2 = - 0-000223. 

64 = 4-000000 

If an increase downwards is reckoned positive, a decrease downwards is to be 
reckoned negative. The first orders of differences are, therefore, positive ; 
the second, negative ; and the third, positive. Substitute a- = in (1) 



= 3-914868 + 0-005407 + 0'000022 + O'OOOOOO. 

= 3-920297. 

The number obtained by simple proportion is 3-920295. The correct number 
is a little greater than 3-920297. 

Lagrange's interpolation formula is more general than the 
above. " Given n consecutive values of a function, to find any 
other intermediate value." 

Let y become t/ , y lt y. 2 , ?/ 3 , . . . , y,, when x becomes a, b, c, 
d, . . . n. The value of y corresponding to any given value of x, 
can be determined from the formula 
y ^(x-b)(x-c). . .(x-n) (x-a)(x-c) . . .(x-n) 

(a-b)(a-c)...(a-n) J ^(b-d)(b-c) . ..(b-nf 1 ^ |(9 
+ (x-a)(x-c) . . . (a-4 
(n-a)(n-b). . . (n-m) '" 



252 HIGHER MATHEMATICS. $ 105. 

If the function is periodic, Gauss' interpolation formula may 
be used. This has a close formal analogy with Lagrange's.* 

= sin|(a - b).sin(x - c) . . . sin^a - n) ,. 

sin^(a - 6).sin(a - c) . . . sm%(a - nf 9 

Lagrange's formula may be employed for the conversion of 
the scale readings of the spectroscope into wave-lengths. Assum- 
ing that the indices of refraction (y , y v y 2 , . . .) are inversely as 
the squares of the wave-lengths (?& , n v n 2 , . . .) if the scale readings 
of, say, three lines near together are given and the wave-lengths of 
two of the lines, the wave-length of the third can be found by 
simple substitution in Lagrange's formula (2), which now assumes 
the form, 



' 



~ 



EXAMPLES. (1) For the three bright magnesium lines, 7 = 5183, 7 2 =5167, 
n = 74'5, n-i = 74-8, n. 2 75 (Lupton). Required the wave-length y^ of the 
third Mg line. 

1 * Oj 2 . 1 03 . = 5173 

y\ ~ (5183) 2 ' 0-5 + (5167) 2 ' 0-5 ' 
Actual measurement gives 5172. 

(2) The scale readings of the Li, Tl and Na lines were found to be re- 
spectively 6-15, 10-55, and 8-0. Required the wave-length of the Na line 
given Li = 6708, Tl = 5351 (Schuster and Lees). Ansr. 5932. 

The most satisfactory method of finding a formula to express 
the relation between the two variables in any set of measurements, 
is to deduce a mathematical expression from known principles or 
laws, and then determine the value of the constants from the ex- 
perimental results themselves. Such expressions are said to be 
theoretical formulae as distinct from empirical formulae, which 
have no well-defined relation with known principles or laws.f 

It is, of course, impossible to determine the correct form of a 
function from the experimental data alone. An infinite number 
of formulae might satisfy the numerical data, in the same sense 

* For the theoretical bases of these reference interpolation formulae the reader 
must consult Boole's work, A T/eatise on the Calculus of Finite Differences, p. 38, 1880. 

f The terms "formula" and "function" are by no means synonymous. The 
formula is not the function, it is only its dress. The fit may or may not be a good 
one. In other words, the function is the relation or law involved in the process. The 
relation may be represented in a formula by symbols which stand for numbers. This 
must be borne in mind when the formal relations of the symbols are made to represent 
some physical process or concrete thing. See the remarks on page 394 with reference 
to the rejection of certain roots of numerical equations. 



S 105. INFINITE SERIES AND THEIR USI> _:,:; 

that an infinite number of curves might be drawn through a series 
of points. (See "Contact of Curves," "Multiple Points," etc.) 
For instance, over thirty empirical formulae have been proposed 
to express the unknown relation between the pressure and tem- 
perature of saturated steam. 

As a matter of fact, empirical formulae frequently originate 
from a lucky guess. Good guessing, here as elsewhere, is a fine 
art. A hint as to the most plausible form of the function is some- 
times obtained by plotting the experimental results. It is useful 
to remember that if the curve increases or decreases regularly, the 
equation is probably algebraic ; if it alternately increases and de- 
creases, the curve is probably expressed by some trignometrical 
function. 

If the curve is a straight line, the equation will be of the form, 
y = mx + b. If not, try y = ax n , or y = ax/(l + bx). If the rate 
of increase (or decrease) of the function is proportional to itself we 
have the compound interest law. In other words, if dy/dx varies 
proportionally with y, y = be~ a * or be ax . If dy/dx varies pro- 
portionally with x/y, try y = bx a . If dy/dx varies as x, try 
y = a + bx 2 . Other general formulae may be tried when the 
above prove unsatisfactory, thus, 

fl \ '7* 

y = T ; y = 10" + bx ; y = a + b log x ; y = a -f be*, etc. 

OC 

Otherwise we may fall back upon Maclaurin's expansion in ascend- 
ing powers of x, the constants being positive, negative or zero. 
This series is particularly useful when the terms converge rapidly, 
96, 2. 

When the results exhibit a periodicity, the general formula to 
be tried, is 

y = a -|- a x sin x + b l cos x + a. 2 sin 2x + b 2 cos 2x + . . . 
If the cycles are regular, only the first three terms on the right 
need be used. Such phenomena are the ebb and flow of tides, 
annual variations of temperature and pressure of the atmosphere, 
cyclic variations in magnetic decimation, etc. See also " Fourier 's. 
Series ". 

Empirical formulae, however closely they agree with facts, do 
not pretend to represent the true relation between the variables 
under consideration. They do little more than follow, more or 
less closely, the course of the graphic curve representing the re- 
lation between the variables within a more or less restricted range. 



254 



HIGHER MATHEMATICS. 



Thus, Eegnault employed three interpolation formulae for the vapour 
pressure of water between - 32 F. and 230 F.* For example, 
from - 32 F. to 0F., he used p = a + ba e ; from to 100 F., 
logp = a + ba e + eft*; from 100 to 230 F., logp = a + bo.* - c(3 . 
Kopp required four formulae to represent his measurements of the 
thermal expansion of water between and 100 C. Each of Kopp's 
formulae was only applicable within the limited range of 25 C. 

Dulong and Petit's memoir, referred to on page 43, is well worth reading 
for some instructive artifices useful in deducing empirical formulae. 

Graphic interpolation. If all attempts to deduce or guess a 
satisfactory formula are unsuccessful, the results are simply tabu- 
lated, or preferably plotted on squared paper, because then "it is 
the thing itself that is before the mind instead of a numerical 
symbol of the thing ". 

Intermediate values may be obtained from the graphic curve 

by measuring the ordinate cor- 
responding to a given abscissa 



IfOO" 














P, 


LAD 


IUM 
> 


^ 














J00' 
IOOO' 
800' 
6OO 
f00 
200 
/r 


















/ 




. 














/ 


















/ 




















/GC 


LD 
<\S,5 


ULP 


HATI 












/ 


FOI 










/ 


















/ 


/ 


















/ 


1EN 
UM 


IUM 
NIU 


BOILS 










/ 


r'Al 


M~" 








zir 


1C*, 


fa 


LPHl 


IR BOILS 








/ 


flE 


on 




















































/WATE 

r i 


I B( 


)ILS 




| Scale Jleaetv 


W 



In measuring high tem- 
peratures by means of the Le 
Chatelier-Austen pyrometer, 
the deflection of the galvano- 
meter index on a millimetre 
scale is caused by the electro- 
motive force generated by the 
heating of a thermo - couple 
(Pt - Pt with 10% Ed) in 
circuit with the galvanometer. 
The displacement of the index 
is nearly proportional to the 
temperature. The scale is cali- 
brated by heating the junction 
to well-defined temperatures 
and plotting the temperatures 
as ordinates, the scale^readings as abscissae. The resulting graph 
or ". .calibration curve " is shown in Fig. KM. The ordinate to 

* Rankine was afterwards lucky enough to find that 

log^ = a - 0/0 - 7/fl 2 , 

represented Regnault's results for the vapour pressure of water throughout the whole 
range - 32 F. to 230 F. 



or vce versa. 



20 fO 60 #0 100 IZ0 JfO 160 ISO 2O0 

FIG. 104. Calibration Chart. 



;i ln<i. INFINITE SERIES AND THEIR I'si-X 

the curve corresponding to any scale reading, gives the desired 
temperature. 

EXAMPLES. (1) What temperature corresponds to a scale reading of 160 
scale divisions in the above diagram ? Ansr. 1300. 

(2) Construct a series of curves from the exposure formula of a thermo- 
meter, 43, (12), between 6 = 0-1 C. and 3-0; x = to x = 300, y = to 
y = 200. What use is the resulting diagram ? 

(3) By plotting on squared paper corresponding values of centimetres 
and inches, litres and pints, grams and ounces, Fahrenheit and Centigrade 
degrees, etc., etc., the mutual conversion of the one into the other can be 
conveniently effected by inspection (i.e., without calculation). Try this : 
given 1 oz. = 28-34 grms., 2 oz. = 56-69 grms., 8 oz. = 226-75 grms., 
1 Ib. = 453-60 grms. 



106. To Evaluate the Constants in Empirical or 
Theoretical Formulae. 

Before a formula containing constants can be adapted to any 
particular process, the numerical values of the constants must be 
accurately known. For instance, the relation 

V = 1 + a.0, 

represents the volume (v) to which unit volume of any gas expands 
when heated to 0. a is a constant. The law embodied in this 
equation can only be applied to a particular gas when a assumes 
the numerical value characteristic of that gas. If we are dealing 
with hydrogen, a - 0-00366 ; if carbon dioxide, a == 0'00371 ; if 
sulphur dioxide, a = 0-00385. 

Again, if we want to apply the law of definite proportions, we 
must know exactly what the definite proportions are before it can 
be decided whether any particular combination is comprised under 
the law. In other words, we must not only know the general law, 
but also particular numbers appropriate to particular elements. 
In mathematical language this means that before a function can 
be used practically, we must know : 

1. The form of the function (i.e., the general formula). 

2. The numerical values of the constants. 

The determination of the form of the function has been discussed 
in the preceding section, the evaluation of the constants remains 
to be considered. 

Is it legitimate to deduce the numerical values of the constants 
from the experiments themselves ? The answer is that the numerical 



256 HIGHER MATHEMATICS. 106. 

data are determined from experiments purposely made by different 
methods under different conditions. When all independently 
furnish the same result it is fair to assume that the experimental 
numbers include the values of the constants under discussion. 
J. F. W. Herschel's A Preliminary Discourse on the Study of 
Natural Philosophy, 221 et seq., is worth reading in this con- 
nection. 

In some determinations of the volume (v) of carbon dioxide 
dissolved in one volume of water at different temperatures (0), the 
following pairs of values were obtained : 

6= 0, 5, 10, 15; 

v = 1-80, 1-45, 1-18, I'OO. 

As Herschel has remarked, in all cases of "direct unimpeded 
action," we may expect the two quantities to vary in a simple 
proportion, so as to obey the linear equation, 

y = a + bx ; we have, v = a + bO, . . (1) 
which, be it observed, is obtained from Maclaurin's series by the 
rejection of all but the first two terms. 

It is required to find from these observations the values of the 
constants, a and b, which will represent the experimental data in 
the best possible manner. 

The above results can be written, 

1. 1-80 = a, \ 

2. 1-45 = a + 56, I 

3. M8-a+ Wb, I 

4. 1-00 = a + 15&J 
which is called a set of observation equations. 

From 

1 and 2, a = 1-80, b = - 0-07, 

2 and 3, a = O64, b = - 0'054, 

3 and 4, a = 0-82, b = - 0-036, etc. 

This want of agreement between the values of the constants 
obtained from different sets of equations is due to errors of 
observation. It nearly always occurs when the attempt is made 
to calculate the constants in this manner. 

The numerical values of the constants deduced from any 
arbitrary set of observation equations can only be absolutely 
correct when the measurements are perfectly accurate. The 
problem here presented is to pick the best representative values 
of the constants from the experimental numbers. If all the 



j; !<><;. INFINITK SKKIKS AND TIIKIIi D8B8, 

measurements were equally trustworthy, the correct method 
\\ould be to find the arithmetical mean of all the values of the 
constants so determined. 

The constants must satisfy the following criterion : The differ- 
ences between the observed and the calculated results must be the 
smallest possible with small positive and negative differences. 

One of the best ways of fixing the numerical values of the 
constants in any formula is to use what is known as the method 
of least squares. This rule proceeds from the assumption that 
the most probable values of the constants are those for which the 
sum of the squares of the differences between the observed and the 
calculated results are the smallest possible (see page 433). 

To take the general case first, let the observed magnitude y 
depend on x in such a way that 

y = a + bx ..... (3) 
It is required to determine the most probable values of a and b. 

For perfect accuracy, we should have the following observation 
equations : 

a -f bx l - y^ = ; a + bx 2 - y% = ; . . . a + bx n - y n = 0. 
In practice this is unattainable. Let v lt v. 2 , . . . v n denote the 
actual deviations so that 

a + bx l - y l = v l ; a + bx. 2 - y. 2 = v 2 ; . . . a + bx n - y n = v n . 
It is required to determine the constants so that, 

2(<y2) = v-f + v.f + . . . + v n 2 is a minimum. 

This condition is fulfilled (page 240) by equating the partial 
derivatives of 2(v 2 ) with respect to a and 6 to zero. In this 
way, we obtain, 

a + bx - y) 2 = 0, hence, 2(a + bx - y) = ; 

^2(a + bx - y)' 2 = 0, hence, 2x(a + bx - y) = 0. 

If there are n observation equations, there are n a's and 3(a) = na, 
therefore, 

na + b$(x) - :%) = ; a^(x) + b$(x 2 ) - ^(xy) = 0. 
Now solve these two simultaneous equations for a and b, 



. . 

[2(z)] 2 - nS(a?) [2(a;)] 2 - n^(x^) ' 

which determines the values of the constants. 

The method of least squares assumes that the observations are 
all equally reliable (see " Errors of Observation," Chapter XL). 

R 



258 



HIGHER MATHEMATICS. 



106. 



Eeturning to the special case at the commencement of this 
section, to find the best representative value of the constants a and 
b in formula (1). 

Previous to substitution in (4), it is best to arrange the data 
according to the following scheme : 



0. 


V. 


*. 


99. 



5 
10 
15 


1-80 
1-45 
1-18 
1-00 



25 
100 
225 



7-25 
11-80 
15-00 


2(0) = 30 2(v) = 5-43 


2(0 2 ) - 350 


2(0r) = 34-05 



Substitute 
observations, 



(4), 



the number of 



these values in equation 
4, hence we get 

a = 1-758 ; b = - 0-0534. 

The amount of gas dissolved at is obtained from the inter- 
polation formula, 

v = 1-758 - 0-05340. 

To show that this is the best possible formula to employ, in 
spite of 1-758 volumes obtained at 0, proceed in the following 
manner : 



Temp. = 6. 


Volume of gas v. 


Difference between 
Calculated and 
Observed. 


Square of Difference 
between Calculated 
and Observed. 


Calculated. 


Observed. 



5 
10 
15 


1-758 
1-491 
1-224 
0-957 


1-80 
1-45 
1-18 
1-00 


- 0-042 
+ 0-041 
+ 0-044 
- 0-043 


0-00176 
0-00168 
0-00194 
0-00185 




0-00723 



The number 0-00723, the sum of the squares of the differences 
between the observed and the calculated results, is a minimum. 
Any alteration in the value of either a or b will cause this term to 
increase. This can easily be verified. For example, if we try the 
very natural a = 1-80, b = - 0*065, we get 0-039; if a = 1-772, 
b = - 0-056 we get 0'0082, etc. (see method of 102). 



$ lot;. INFIMTK SKUIES AND THEIR USI-X L'.V.) 

Si Xntiir,; 63, 489, 1901. The above method of treatment IB founded on 
that of Kohlrausch in his Leitfaden der praktisdu-n I'lujxik (Teubner, Leipzig, 
1896), p. 8. For other methods of calculating the constants, see Lupton's 
.Wrs on Observations, p. 105, 1898 ; and 186. 

KXVMPLES. (1) Find the law connecting the length (/) of a rod with 
temperature (0), when the length of a metre bar at elongates with rise of 
temperature according to the following scheme : 

= 20, 48, 50, 60 C. ; 

1 = 1000-22, 1000-65, 1000-90, 1001-05 mm. 

(Kohlrausch, I.e.). During the calculation, for the sake of brevity, use 
/ = -22, -65, -9 and 1-05. Assume I = a + be and show that a = 999-804, 
b = 0-0212. 

(2) Find a formula similar to (4) for the general equation y = a tan a + 6, 
where a and b are constants to be determined. 

(3) According to Bremer's measurements aqueous solutions of sodium 
carbonate containing p / of the salt expand by an amount v as indicated in 
the following table : 

p = 3-2420, 4-8122, 7'4587, 10-1400 ; 
10 4 x v = 1-766, 2-046, 2-342, 2-732. 

Hence show that if v = a + bp, a = 0-00012415, b = 0-00001528. 

Suppose that instead of the general formula (3), we had 
started with 

y = a + bx + ex 2 , . . . (5) 

where a, b and c are constants to be determined. The resulting 
formulae for b and c (omitting a), analogous to (4), are, 

These two formulae have been deduced by a similar method to 
that employed in the preceding case, a is a constant to be 
determined separately by arranging the experiment so that when 
x = 0, a = y Q . 

EXAMPLES. (1) The following observations were made by Bremer. If p 
denotes the density of an aqueous solution of calcium chloride at U C., 



9. p. 

15-65 1-03336 

20-11 1-03273 

28-60 1-02856 



e. p. I e. p. 



33-40 1-02356 
39-25 1-02640 
46-01 1-02348 



32-76 1-02051 
63-23 1-01516 



Calculate the constants a and b in the formula, 

p = Po (l + ae + btf*), 
where Po = 1-03619. Ansr. b = - 0-000003301 ; a = - 0-0001126. 

(2) The following series of measurements of the temperature (e) at different 
depths (x) in an artesian well, were made at Grenelle (France) : 



260 HIGHER MATHEMATICS. 106. 

x = 28, 66, 173, 248, 298, 400, 505, 548 ; 

= 11-71, 12-90, 16-40, 20-00, 22-20, 23-75, 26-45, 27-70. 
The mean temperature at the surface was 10-6. Hence show that at a depth 
of x metres, the temperature will be, 

e = 10-6 + 0-042096x - 0-000020558^ 2 . 
(3) If, when x = 0, y = 1 and when 

x = 8-97, 20-56, 36-10, 49-96, 62-38, 83'73 ; 

y = 1-0078, 1-0184, 1-0317 1-0443, 1-0563, 1-0759. 

Hence show that 

y = 1 + 0-00084.K + 0-0000009z 2 . 

The reader will himself have to deduce the general formulae 
for a, b, c, when still another correction term is included, namely, 

y = ax + bx 2 + ex 3 . . . (8) 

EXAMPLES. The following measurements are selected from a paper by 
Thompson in Wiedemann's Annalen (44, 553, 1891). 

(1) If when 

x = 0-2, 0-4, 0-6, 0-8, 1-0, 1-2, 

p = 5-531, 11-084, 16-671, 22-298, 27*949, 33-646, 
show that 

x = 27-578p + 0-3193p 2 + 0'0538p 3 . 

(2) If when 

x = 0-2, 0-4, 0-6, 0-8, 1-0, 

p = 7-078, 14-196, 21-358, 28-558, 35-792, 
show that 

x = 35-2725p + 0-5725p 2 - 0'0525p 3 . 

If three variables are to be investigated, we may use the 
general formula 

z = ax + by (9) 

The reader may be able to prove, on the above lines, that 

A rough and ready method for calculating the constants is to 
pick out as many observation equations as there are unknowns 
and solve for x, y, z, by ordinary a, b, c, say, algebraic methods. 
The different values of the unknown corresponding to the different 
sets of observation arbitrarily selected are thus ignored. 

EXAMPLE. Corresponding values of the variables x and y are known, say, 
#!> 2/i 5 X 2> 2/2 5 x s> 2/3 J Calculate the constants a, b, c, in the interpolation 
formula 

y * o(10)*+1. 



$ KM;. INFINITE si-:uiK> AND TIIKIlt CJ8E8. x\\ 

When Xj = 0, .Vi = . Thus b and c remain to be determined. Take logarithms 
of the two equations in x 2 , y. 2 and a- 3 , y 3 and show that, 



This method may be used with any of the above formulae when 
an exact determination of the constants is of no particular interest, 
or when the errors of observation are relatively small. 

Graphic Method. Returning to the solubility determinations 
at the beginning of this section, prick points corresponding to 
pairs of values of v and 6 on squared paper. The points lie 
approximately on a straight line. Stretch a black thread so as to 
get the straight line which lies most evenly among the points. 
Two points lying on the black thread line are v = 1*0, = 14-5, 
and v = 1-7, B = 1-5, 

.-. a + 14-56 = 1; a + 1-56 = 1-7. 
By subtraction, b = - 0'54, .-. a = 1-78. 

It is here assumed that the curve which goes most evenly 
among the points represents the correct law (footnote, page 123). 
In the example just considered, there is, perhaps, too small a 
number of observations to show the method to advantage. Try 
these : 

p = 2, 4, 6, 8, 10, 20, 25, 30, 35, 40, 
s = 1-02, 1-03, 1-06, 1-07, 1-09, 1-18, 1-23, 1-29, 1-34, 1-40, 
where s denotes the density of aqueous solutions containing p / 
of calcium chloride at 15 C. The selection of the best "black 
thread " line is more uncertain the greater the magnitude of the 
errors of observation affecting the measurements. The values 
deduced for the constants will differ slightly with different 
workers or even with the same worker at different times. With 
care, and accurately ruled paper, the results are sufficiently accurate 
for most practical requirements. 

When the " best " curve has to be drawn freehand, the results 
are more uncertain. For example, the amount of " active " oxy- 
gen (y) contained in a solution of hydrogen dioxide in dilute 
sulphuric acid was found, after the elapse of t days, to be : 

t = 6, 9, 10, 14, 18, 27, 34, 38, 41, 54, 87, 

y = 3-4, 3-1, 3-1, 2-6, 2-2, 1-3, 0-9, 0'7, 0-6, 0-4, 0-2, 



262 HIGHER MATHEMATICS. $ 106. 

where y = 3 '9 when t = 0. We leave these measurements with 
the reader as an exercise. 

In Perry's Practical Mathematics (published by the Science and 
Art Department, London, 1899, 6d.), a trial plotting on "logar- 
ithmic paper" is recommended in certain cases. On squared 
paper, the distances between the horizontal and vertical lines are 
in fractions of a metre or of a foot. On logarithmic paper, the 
distances between the lines are proportional to the logarithms of 
the numbers. If, therefore, the experimental numbers follow a 
law like 

Iog 10 # + alog 10 2/ = constant, 

the function can be plotted as easily as on squared paper. If the 
resulting graph is a straight line, we may be sure that we are 
dealing with some such law as 

xy a = constant ; or, (x + a) (y + b) a = constant. 

EXAMPLE. The pressure (p) of saturated steam in pounds per square 
inch when the volume is v cubic feet per pound is 

p = 10, 20, 30, 40, 50, 60, 

v = 37-80, 19-72, 13-48, 10-29, 8-34, 6-62. 
(Gray's Smithsonian Physical Tables, 1896.) Hence, by plotting correspond- 
ing values of p and v on logarithmic paper, we get the straight line : 

lo gio-P + lo gio^ = lo gio& ; hence, po 1 * = 382, 
since Iog 10 6 = 2-5811, .-. b = 382 and a = 1-065. 

Logarithmic paper is not difficult to make. The gradations on 
the slide rule give the correct distances without calculation. 

A semi-logarithmic paper may be made with distances be- 
tween say the vertical columns in fractions of a metre, while the 
distances between the horizontal columns are proportional to the 
logarithms of the numbers. Functions obeying the compound 
interest law will plot, on such paper, as a straight line. One 
advantage of logarithmic papers is that the skill required for 
drawing an accurate freehand curve is not required. The stretched 
black thread will be found sufficient. With semi-logarithmic paper, 
either 

x + Iog 10 2/ = constant ; or, y + Iog 10 a; = constant 
will give a straight line. 

EXAMPLES. (1) Plot on semi-logarithmic paper Harcourt and Esson's 
numbers (I.e.) : 

t = 2, 5, 8, 11, 14, 17, 27, 31, 35, 44, 

y = 94-8, 87-9, 81-3, 74-9, 68-7, 64-0, 49-3, 44-0, 39-1, 31-6, 



;i 1<>7. INFIMTK SERIES AND THEIR USES. 263 

for the amount of substance y remaining in a reacting system after the elapse 
of an interval of time t. Hence determine values for the constants a and 6 in 

y = ae~ M , i.e., in Iog 10 7/ + bt = Iog 10 a, 
a straight line on " semi-log " paper. 

(2) What " law " can you find in Perry's numbers (Proc. Roy. Soc., 23, 
472, 1875), 

6 = 58, 86, 148, 166, 188, 202, 210, 
C = 0, -004, -018, -029, -051, -073, -090, 
for the electrical conductivity C of glass at a temperature of 6 F. ? 

(3) Evaluate the constant a in Arrhenius' formula, 77 = a*, for the viscosity 
17 of an aqueous solution of sodium benzoate of concentration x, given 

n = 1-6498, 1-2780, 1-1303, 1-0623, 



107. Approximate Integration. 

We have seen that the area enclosed by a curve can be esti- 
mated by finding the value of a definite integral. This operation 
may be reversed. The numerical value of a definite integral can be 
determined from measurements of the area enclosed by the curve. 
For instance, if the integral \f(x) . dx is unknown, the value of 

I f(x) . dx can be found by plotting the curve y = f(x), erecting 

ordinates to the curve on the points x = a and x = b and then 
measuring the surface bounded by the #-axis, the two ordinates 
just drawn and the curve itself. 

This area may be measured by means of the planimeter, an 
instrument which automatically registers the area of any plane 
figure when a tracer is passed round the boundary lines.* 

Another way is to cut the figure out of a sheet of paper, or 
other uniform material. Let w 1 be the weight of a known area a^ 
and w the weight of the piece cut out. The desired area x can 
then be obtained by simple proportion, 

w l : a = w : x. 

Interpolation formulae may be used for the approximate evalu- 
ation of any integral between certain limits. The problem may 
be stated thus : Divide the curve into n portions bounded by 
n + 1 equidistant ordinates y , y v y. 2 , . . ., y,,, whose magnitude 
and common distance apart is known, it is required to find an 



* A good description of these instruments will be found in the British Association's 
Reports for 1894, page 496. 



264 



HIGHER MATHEMATICS. 



107. 



approximate expression for the area so divided, that is to say, to 
evaluate the integral 

f "f(x) . dx. 
Jo 
Assuming Newton's interpolation formula we may write, 

f(x) = 2/ + x\ + x(x - 1)A* + (1) 

) . dx = yTdx + *\[x . dx + [$x(x - !)<** + . . ., (2) 
o Jo Jo Jo^ : 

which is known as the Newton-Cotes integration formula. We 
may now apply this to special cases, such as calculating the value 
of a definite integral from a set of experimental measurements, etc. 

1. Parabolic Formulae. Take three ordinates. Eeject all terms 
after A 2 . Eemember that A^ = y l - y and A 2 = y. 2 - fy l + y . 
Let the common difference be unity, 



f(x) . dx = 22/ + 2A 1 + |-A 2 = l(y Q + ^y l + y. 2 ). (3) 

If h represents the common distance of the ordinates apart, we 
have the familiar result known as Simpson's one-third rule, thus, 

.dx = 



A graphic representation will perhaps make the assumptions in- 

volved in this formula 
more apparent. 

Make the construc- 
tion shown in Fig. 105. 
We seek the area of 
the portion ANN' A' 
corresponding to the 
integral f(x) . dx be- 
tween the limits x = x$ 
FIG. 105. Approximate Integration. and x = x llt where f(x) 

represents the equation to the curve ABC . . . MN. 

Assume that each strip is bounded on one side by a parabolic curve. 
Area CDEE'C' = Area of trapezium CEE'C' + Area parabolic 

segment CED. 
From well-known mensuration formulae (15), page 491, 

CDEE'C' = C'E'[(CC' + EE') + %{DD' - $(CC' + EE')}] ; 



(5) 




h(CC' + DD' + EE') 



.< 107. INFINITE s|-:i:il-> AND THEIR USI-> -jr.:, 

Extend this discussion to include the whole figure, 

Area ANN' A' = J/i(l + 4 + 2 + 4 + . . . + 2 + 4 + 1), (6) 
\\hore the successive coefficients of the perpendiculars A A', BB', . . . 
alone are stated ; h represents the distance of the strips apart. The 
greater the number of equal parts into which the area is divided, 
the more closely will the calculated correspond with true area. 

Put OA' = x ; ON = x n ; A'N' = x n - X Q and divide the area 
into n parts ; h = (x n - x )/n. Let t/ , y lt y 2 , . . . y n denote the 
successive ordinates erected upon Ox, then equation (6) may be 
written in the form, 

J*n 
Jftx) dx = *h{(y + y,,) + ifat + y z + . . . + y n _ ,) | ^ ^ 

+ %2 + 2/ 4 + - - - +y-i) I 

In practical work a great deal of trouble is avoided by making 
the measurements at equal intervals x l - X Q , x 2 - x l , . . ., x n -x n _*. 

EXAMPLE. In measuring the magnitude of an electric current by means 
of the hydrogen voltameter, let C , C lt C 2 , . . . denote the currents passing 
through the galvanometer at the times t Q , t lt t 2 , . . . minutes. The volume of 
hydrogen liberated (v) will be equal to the product of the intensity of electri- 
city (C amperes), the time (t), and the electrochemical equivalent of the 
hydrogen x, (v = xCt). 

Arrange the observations so that the galvanometer is read after the elapse 
of equal intervals of time. Hence ^ - ? = t 2 - ^ = t z - t z = . . . = h. 
From (7), 



C,+ . . . + C_ 1 ) + 2(C 8 +C 4 + . . . +C n _ 2 ){, 
In an experiment, v = 0-22 when t = 3, and 

t = 1-0, 1-5, 2-0, 2-5, 3-0, . . . ; 

C = 1-53, 1-03, 0-90, 0-84, 0-57, . . . 
' 0*5 

C . dt = -g-{(l-58 + 0-57) + 4(1-03 + 0-84) + 2 x 0-90} = 1-897. 

.-. x = -22/1-897 = 0-1159. 

This example also illustrates how the value of an integral can be obtained 
from a table of numerical measurements. 

The result 0-1159, is better than if we had simply proceeded by what 
appears, at first sight, the correct method (see (13) below), namely, 

'.#- ft - < )QL+^i + ( / a - t^^ + . . . = i-9i, 

for then x = -22/1-91 = 0-1152. 

The correct value is 0-116 nearly. 

If we take four instead of the three ordinates in the preceding 
discussion, we obtain 



266 



HIGHER MATHEMATICS. 



107. 



where h denotes the distance of the ordinates apart, y Qt y v . . . 
the ordinates of the successive perpendiculars in the preceding 
diagram. This formula is known as Simpson's three-eighths 
rule. 

If five, six or seven ordinates are taken, the corresponding 
formulae are respectively 

(&) - dx = &h(7y + 32 2/1 + 12y 2 + 32y 8 + 7y 4 ). . 



\' 

Jo 



f(x) . dx 



. dx = 



(9) 
(10) 
(11) 



The last result, known as Weddle's rule, is said to give very 
accurate results in mensuration problems. 

All these formulae are discussed in Boole's Calculus of Finite Differences 
(I.e.] under the heading " Mechanical Quadrature ". 

EXAMPLE. Evaluate the integral fxP.dx between the limits 1 and 11 by 
the aid of formula (6), given h 1 and y , y lt y 2 , y 3 , . . . y 8 , y 9 , y w are respec- 
tively 1, 8, 27, 64, . . . , 1000, 1331. Compare the result with the absolutely 
correct value. From (6), 

TV . dx = (10970) = 3656-|. 
By actual integration, the perfect result is, 



- i(l) 4 = 3660. 

The reader will perhaps have met some of the above formulae in his 
arithmetic (mensuration). 

2. Trapezoidal Formulae. Instead of assuming each strip to 
y M N be the sum of a trapezium and a 

parabolic segment, we may s^ty>P se 
that each strip is a complete tra- 
pezium. In Fig. 106, let AN be a 
curve whose equation is y = f(x) ; 
A A', BB', . . . perpendiculars 
drawn from the re-axis. The area 
of the portion ANNA' is to be 
Let OB' - OA' = OC' - OB' = . . = h. It follows 




C fr 

FIG. 106. 



determined. 

from known mensuration formulae, (10), page 491, 

Area ANNA' = \h(AA' + BB') + . . . + \(MM' + NN), 

= h(AA'+ 1BB'= 2CC'+ . . . + VMM' + NN') t 
= fc(i + i + i + . . . + l + 1 + i), . (12) 



< I os. INFINITE SERIES AND THKIi; l-l> w\7 

where the coefficients of the successive ordinates alone are written. 
This result is known as the trapezoidal rule. 

Let x , x v x. 2 , . . . , x n , be the values of the abscissae corre- 
sponding to the ordinates f/ , y lt y 2 , . . . , y n , then, 



If o?j - x = x. 2 - x l = . . . = h, we get, by multiplying out, 
C X H 

J* ^ 

The trapezoidal rule, though more easily manipulated, is less 
accurate than those based on the parabolic formula of Newton and 
Cotes. 

The following expression, 

Area ANN' A' = h(^ + if + 1 + 1 + . . . + 1 + 1 + if + ^), (15) 
or, 



f* 

ft r>\ 
I /\*/' 

J^n 



. . . +y n -^ (16) 



is said to combine the accuracy of the parabolic rule with the 
simplicity of the trapezoidal. It is called Durand's rule. 

EXAMPLE. Evaluate the integral / , by the approximate formulae 

Jz x 

(7), (14) and (16), assuming h = 1, n = 8. Find the absolute value of the 
result and show that these approximation formulae give more accurate 
results when the interval h is made smaller. Ansr. (7) gives 1-611, (12) 
gives 1-629, (15) gives 1-616. The correct result is 1-610. 

Lemoine (Annales de Chimie et de Physique [4], 27, 289, 1872) encountered 
some non-integrable equations during his study of the action of heat on red 
phosphorus. In consequence, he adopted these methods of approximation. 
The resulting tables "calculated" and "observed" were very satisfactory. 
For these, see the cited memoir. 

Another method of approximate integration, of special import- 
ance in practical work, will now be indicated. 

108. Integration by Infinite Series. 

It is a very common thing to find expressions not integrable 
by the ordinary methods at our disposal. We may then resort 
to the methods of the preceding section, or, if the integral can 
be expanded in the form of a converging series of ascending 
or descending powers of x, we can integrate each term of the 



268 HIGHER MATHEMATICS. $ 108. 

expanded series separately and thus obtain any desired degree .of 
accuracy by summing up a finite number of these terms. 

If f(x) can be developed into a converging series, f(x) . dx is also 
convergent. Thus if 

f(x) = 1 + x + x 2 + x' 3 + . . . +x n ~ l + x 11 + . . . (1) 

\f(x).dx = x + }rf + \x*+ . . . +ir"+- x n+l + ... (2) 

2i O Hj Tl -f- 1 

Series (1) is convergent when x is less than unity, for all values of 
n. Series (2) is convergent when nx/(n + 1) and therefore when 
x is less than unity. The convergency of the two series thus 
depends on the same condition. If the one is convergent, the 
other must be the same. 

If the reader is able to develop a function in terms of Taylor's 
series, this method of integration will require but few words of 
explanation. One illustration will suffice. 

By division, or by Taylor's theorem, 



\ 

f ^ 

' ' \ 1 | ^y.2 

J 1 + X 



jic* j . dx -\- ^x . dx \x . dx 

x - x 3 + -x 5 - . . . = tan - l x. 

o O 



page 229, (6). 



EXAMPLES. (1) Using the approximation of Simpson, (7) preceding sec- 
tion, show that 

C dx = tan _ 12 _ tan-i! = 0-821751. 
j I + x* 

Verify the following results. 




(7) How would you propose to integrate J(log 10 a; . dx)l(l - a-) in series ? 
See also pages 188 and 355. 

(8) The result of the following discussion is required later on. To find a 
value for the integral 



$ 108. INFINITE SERIES AND THKII; IH> j;-j 

Integrals of this type are extensively employed in the solution of physical 
problems. A'.;/., in the investigation of the path of a ray of light through the 
atmosphere (Kramp) ; the conduction of heat (Fourier) ; the secular cooling of 
the earth (Kelvin), etc. One solution of the important differential equation 



is represented by this integral. See also Chapter VIII., Jj 152 and 154. On 
account of its paramount importance in the theory of errors of observation 
(</. c.), (3) is sometimes called the error function, and written " erf x ". 

Glaisher (Phil. Mag. [4], 42, 294, 421, 1871) and Pendlebury (ib., p. 437) 
have given a list of integrals expressible in terms of the error function. The 
numerical value of any integral which can be reduced to the error function, 
may then be read off directly from known tables. See Chapter XI., 180 
also Burgess, Trans. Roy. Soc. Edin., 39, 257, 1898. 

NOTE. The error function (3) may be expressed as a gamma function, 
^r(^), or ^V'T, from (12), 83. 

The following ingenious method of integration is due to Gauss : If a sur- 
face has the equation 

z = e-( x * + ^), ..... (4) 

the volume included between this surface, the .z-plane (for which z = 0), the 
x-plane (between the limits x = and x = oo) and the y-plane (between the 
limits and oo), is given by the expression, 

volume 



= I I e-^ + v^dx.dy = / e~^dxl e~y' 2 dy. . (5) 

J J .'0 Jo 



Let u denote the integral of the original equation, (3), then, it follows that 
the volume in (5) = u 2 . 

Again, if we express z in polar coordinates, since x? + y 2 = r 2 , z = e-* 2 , 
the area of an element in the 2-plane becomes r.de.dr, instead of dy.dx. 
In order that the limits may extend over the same part of the solid as before, 
the integration limits must be transformed so that r extends over and oo 
and 6 over and fa. Therefore the volume of the solid in polar coordinates, is 

volume = / / e~ ^r.de.dr. 

Jo Jo 
Integrate with respect to and 

volume = fa I e~^r . dr. 
Now multiply and divide by - 2- and integrate. 

.. volume = - ^ir I e ~ r<i J = Jr. 

.-. w 2 = Jir ; or, u = \ \V. 
By successive reduction ( 75), 

-**.4,-<>'-W-_* >.-*..*,. . ,6) 

when n is odd, and 



when n is even. 



270 HIGHER MATHEMATICS. 108. 

All these integrals are of considerable importance in the kinetic theory 
of gases and in the theory of probability. Common integrals in the former 
theory are 



Vir Jo \>irJo ._ ^.^ . . 

From (7), the first one may be written %Nma?, the latter, JZNal \/TT. 

If the limits are finite, as, for instance, in the probability integral, 

P = /= I e - h2 ^d(hx). 

VirJ 
Put hx = t, then 



Develop e - f2 into a series by Maclaurin's theorem, as just done in example (4) 
above. The result is that 
2 



may be used for small values of t. 

For large values, integrate by parts, 






1 -< 



From (4), page 185, 

l'e- a dt = fe-^dt - r*-*dt. 

Jo Jo J t _ 

The first integral on the right-hand side = ^N/TT. Integrating the second 
between the limits oo and t 

p-'i--^S(i-i-+pJ-2y^+...). - . (io) 

This series converges rapidly for large values of t. From this expression the 
value of P can be found with any desired degree of accuracy. 



PART II. 

ADVANCED. 

CHAPTER VI. 

HYPERBOLIC FUNCTIONS. 

109. Euler's Exponential Values of the Sine and Cosine. 

THERE are certain combinations of the exponential functions which 
are frequently employed in the various branches of physics. These 
functions bear the same formal resemblance to one half of a rect- 
angular hyperbola that the circular functions of trignometry do to 
the circle, hence their name hyperbolic functions. 

Hyberbolic functions have now become so incorporated with 
practical formulae that it is necessary to have at least an elementary 
knowledge of their properties. 

Returning to the imaginary *J - 1, i has no physical meaning, 
it is an abstract mathematical concept to which mathematicians 
have arbitrarily applied the fundamental laws of algebra distri- 
butive law, commutative (" relatively free ") law, and the index 
law. See footnotes, pages 175 and 304. 

In 98, (9) and (11), the following series were developed : 

x 2 x s x 2 x* 

e" = 1 +# + 2T+3~i + ' 5 e ~* = l ~ x + 21 ~3~[ + - C 1 ) 

If we substitute ix in place of x (see footnote, page 175), we obtain, 
LX x 1 ix"' x 4 tx r 

-T"2T"3T + ^! + 5T--'- ; 

x 2 x* \ (x x s x b 

- 2T + 4T - ) + <T - 3T+5T - ' 
By reference to page 229, we shall find that the first expression in 
brackets, is the cosine series, the second, the sine series. Hence, 
e 1 -'' = cos x + L sin x. ... (3) 



272 HIGHER MATHEMATICS. 110. 

In the same way, it can be shown that 



X 1 



Or, e ~ LX = cos x - i sin x. . . . (5) 

Combining equations (3) and (5), 

1(01* - e - **) = i sin a; ; (e ix + e~ LX ) = cos x. . (6) 



110. The Derivation of Hyperbolic Functions. 

Every point on the perimeter of a circle is equidistant from the 
centre, therefore, the radius of any given circle has a constant 
magnitude, whatever portion of the arc be taken. 

In plane trignometry, an angle is conveniently measured as a 
function of the arc of a circle. Thus, if /' denotes the length of 
an arc of a circle subtending an angle 6 at the centre, r' the radius 
of the circle, then 

arc _ l_ 
~ radius r'' 

This is called the circular measure of an angle and, for this reason, 
trignometrical functions are sometimes called circular functions. 

This property is possessed by no plane curve other than the 
circle. For instance, the hyperbola, though symmetrically placed 
with respect to its centre, is not at all points equidistant from it. 
The same thing is true of the ellipse. The parabola has no centre. 

If I denotes the length of the arc of any hyperbola which cuts 
the re-axis at a distance r from the centre, the ratio 

I 

u = -, 
r 

is called an hyperbolic function of u, just as the ratio l'/r' is a 
circular function of 6. 

To find a value for the ratio u = l/r. For the rectangular 
hyperbola 

y = J(tf> - a 2 ) ; .-. dy/dx = x/ J(x* - a 2 ). . (1) 
The length of any small portion dl of the arc of an hyperbola is, 
by 81, 

/ //7/\ 2 . /Q/y2 _ /,2 

dl = 



110. HYPERBOLIC FUNCTION- OT 

The distance r of any point (x, y) from the origin on this curve, is 



- a 2 

If x l is the abscissa of any point on the hyperbola, a the 
abscissa of the starting point, 



_ 
- 



IX" a, a, 

Put x for x l and, remembering that log e e = 1, 

u log e = log (x + V# 2 - a 2 )/a ; a . . 

or, e u = (a; 




* .**, 



.-. (e u - z/a) 2 = aW - 1, or 2xe u /a = e 2 " + 1 ; 



. ?=^u + e -) V ^ 

' CL 

But this relation is practically that developed for cos x, (6), of 
the preceding section, ix, of course, being written for u. The ratio 
x/a is defined as the hyperbolic cosine of u. It is usually written 
cosh u and pronounced " cosh u" or " h-cosine u ". Hence, 

cosh u = ^(e u + e~ w ) = l+^ + j-j + ... (2) 

In the same way, proceeding from (1), it can be shown that 
y = Ix* _ _ /e 2 " + 2 + e - 2u _ _ /e 2M - 2 + e - 



a relation previously developed for i sin aj. The ratio y/a is called 
the hyperbolic sine of u, written sinh u, pronounced " shin u," 
or " h-sine u ". As before 

n 1 3 At 5 

sinh ?t = J(e M -e- M ) = w+^y + ^y+... (3) 

The remaining four hyperbolic functions, analogous to the 
remaining four trignometrical functions, are tanh u (pronounced 
" h-tan u," or "tank w"), cosech u, sech u and coth u. Values for 
each of these functions may be deduced from their relations with 
sinh u and cosh u. Thus, 

sinh u 1 

tanh u = r ; sech u = 



cosh u ' cosh u ' \ 



coth u = . ; cosech u 



tanh u ' sinh 

Unlike the circular functions, the ratios x/a, y/a, when referred 
to the hyperbola, do not represent angles. An hyperbolic function 

S 



274 



HIGHER MATHEMATICS. 



ill. 




expresses a certain relation between the coordinates of a given point 
on the arc of a rectangular hyperbola. 

Let (Fig. 107) be the centre 
of the hyperbola APB, described 
about the coordinate axes Ox, Oy. 
From any point P(x, y) drop a per- 
pendicular PM on to the #-axis. 
Let OM = x, PM = y, OA = a. 

.'. coshu = xfa; sinhu = y/a. 
For the rectangular hyperbola, 
FIG. 107. x <2 - y* = a 2 . 

.. a 2 sinh 2 w - o. 2 cosh% = a 2 ; or, sinh% - cosh 2 ^ = 1. 
The last formula thus resembles the well-known, 

cos' 2 x + sin 2 # = 1. 

Draw P'M a tangent to the circle AP' at P. Drop a perpendicular 
PM' on to the ic-axis. Let the angle M'OP = 0. 

.-. x/a = sec0 = coshw ; y/a = tan# = sinhw. . (5) 

In example (5)., page 279, it is shown that the area AOP' = ^a 2 

and of A OP = \cPu. From equations, page 273, it follows that 

e u = cosh u 4- sinh u = sec + tan 0. 

u = log(sec(9 + tan0) = Iogtan(j7r + 0), . (6) 

and tanh \u = tan |0. . . . (7) 

When is connected with u by any of the four relations (5), 

(6) and (7), 6 is said to be the Gudermannian of u and written 

= gd%. The Gudermannian function, therefore, connects the 

circular with the hyperbolic functions. 



111. The Graphic Representation of the Hyperbolic 
Functions. 

We have seen that the trignometrical sine, cosine, etc., are 
periodic functions. The hyperbolic functions are exponential, not 
periodic.* This will be evident from the following diagrams (Figs. 

* Since cos x and sin x are periodic functions, cos x + i sin x repeats its value every 
time x is increased by 2w ; it therefore follows that e lx also repeats its value every time 
x is increased by 2ir. In this particular case e ix is said to be an imaginary iwrfoitii: 
function of x. 

To illustrate the periodic nature of the symbol i, suppose \/ - 1 represents the 
symbol of an operation which when repeated twice changes the sign of the subject of 
the operation, and when repeated four times restores the subject of the operation to 
its original form. For instance, if we twice operate on x with s/ - 1, we get - x, or 



111. 



HYN-llinoLIC FUNCTIONS, 



108, 109, 110), which represent graphs of the six hyperbolic 
functions. 






V 

108. Graphs of tanh x and 
coth x. 



FIG. 109. Graphs of cosh x and 
sech x. 



(\'~i )* = *, 

and so on in cycles of four. If the imaginary quantities o:, - ix, . . . are plotted on 
the y-axis (axis of imaginaries), and the real quantities x, - x, ... on the avaxis 
(d.ri* <>f reals), the operation of \/ - 1 on x will rotate x through 90, two operations 
will rotate x through 180, three operations will rotate x through 270, and four 
operations will carry x back to its original position. 
Since 2i sin x = e lX - e ~ iar , if x = v, sin v = 0, 

. . e l1t - e ~ Llr = ; or, <e t7r = e ~ t7r , 

meaning that the function e tx has the same value when x = ir and when x = - v. From 
the last equation, 



But 



= g log a; + 



which means that the addition of 2tir to the logarithm of any quantity has the effect 
of multiplying it by unity, and will not change its value. Every real quantity, there- 
f ,->>, has one real logarithm and an infinite number of imaginary logarithms differing 
by 2j/r, where n is an integer. 

When any function has two or more values for any assigned real or imaginary 
value of the independent variable, it is said to be a multiple-valued function. Such 
are logarithmic, irrational algebraic, and inverse trignometrical functions. The 
imaginary values in no way interfere with the ordinary arithmetical ones. A single- 
valued function assumes one single value for any assigned (real or imaginary) value of 
tin- independent variable. For example, rational algebraic, exponential and trigno- 
metrical functions are single-valued functions. 

There are several interesting relations between sin x and e*. Thus, if 
y = a sin qt + b sin qt, (Py/dP =-q*y, dy*/dt* = q*y ; 
y = et, d*y/dt* = q*y ; iPy/dP = q*y, etc. 



276 



HIGHER MATHEMATICS. 



112. 



The graph 

y = cosh re, 

is known in statics as the "catenary". Tanhsc and cothrc have 
an imaginary period TTI, the remaining hyperbolic functions have 
the imaginary period 2?. 







FIG. 110. Graphs of sinh x and cosecha;. 

112. Transformation and Conversion Formulae. 

(i.) To pass from trignometrical to hyperbolic functions and vice 
versa. By substituting J 1 . x, or, ix in place of u in equations 
(2) and (3), 110, we obtain 

cosx. . . (1) 



Or, 



cosh ix 


Li._? 


2 x 4 


2! ' 4! 


sinhiic 




\ ~4~ 


31 5! 


cos a? = 


%(e lx + 


e~ LX ) = cosh tic. 




1 


i 


sinz = 


if - 


e~ 1 *) = -sinhi; 
t 



(2) 
(3) 



This set of formulae allows the trignometrical and hyperbolic 

functions to be mutually converted into each other. 

(ii.) Conversion formulae. Corresponding to the trignometrical 

formulae there is a great number of relations among the hyperbolic 

functions, such as 

cosh 2 # - sinh 2 z = 1. ..... (5) 

cosh2ic= 1 + 2sinh 2 # = 2cosh 2 # + 1. . . (6) 

sinh x - sinh y = 2 cosh J(# + y) . sinh ^(x - y), (7) 

and so on. These have been summarised in the chapter, " Collection 

of Reference Formulae ". 



EXAMPLE. Show 



tanh ix = i tan x. 



114. HYPERBOLIC FUNCTION- 877 

113. Inverse Hyperbolic Functions. 

The inverse hyperbolic functions are denned in the same way 
as the inverse trignometrical functions, that is to say, 

smh- l y = x, 
is another way of stating that 

y = sinh re. 

These inverse functions can be expressed as logarithmic func- 
tions, since, 

y = sinha? = ^(e x - e~ r ), 

.-. &* - fye* -1 = 0. 
Solve as a quadratic. 

.-. * = y Vy 2 + l. 

For real values of x, the negative sign is excluded in the case of 
sinh l y, and 

*/ = log{y + VyTT>. (!) 



Similarly cosh~ l y = \og{y Jy* - 1} ; (2) 

Here (Fig. 109) we can use either value. 

+ y)/(l - y) ; . . (3) 

+ l)f(y - 1) ; . . (4) 

= log {1 + x/1 - y*\ly ; . . (5) 

cosech~ l y = log {1 + >/l -f- y 2 \/y. . . (6) 

114. Differentiation and Integration of the Hyperbolic 

Functions. 

The functions may be differentiated in a similar manner to the 
ordinary trignometrical functions. The symbol V - 1 is treated 
as if it were a constant real quantity. Thus, let 

y = sinhrr =f(x), -'.f(x -f h) = sinh(# + h). 
dy _ j-. sinh(ic + h) - ainhh 

T . 2 sinh $h . cosh(ic + $h) 



The limit of amhu/u when u = 0, is unity (page 505), just as in 
the somewhat analogous sin BIB = 1, when 6 becomes vanishingly 
small. 

.-. dy/dx = d(smhx)/dx = cosh a;. 



278 



HIGHER MATHEMATICS. 



114. 



This is proved more directly as follows : 

d(smhx)/dx = d{^(e x - e~ x )}/dx 

= \(e x + e~ x ) = coshic. 
For the inverse hyperbolic functions, let 

y = sinh l x, 
.-. dx/dy = coshy. 
From (5), 112, 

coshy = v/sinh 2 2/ + 1 ; .-. coshy = -Jx 2 + 1, 
from the original function to be differentiated. 

.. dy/dx = I/ \/# 2 + 1. 

EXAMPLE. If y = cosh mx + sinh mx, show that 
d 2 y/dx z = m 2 y. 

A standard collection of results of the differentiation and inte- 
gration of hyperbolic functions, is set forth in the following table: 

TABLE III. STANDARD INTEGRALS. 



Function. 


Differential Calculus. 


Integral Calculus. 


y = sinh x. 


-<co8ha. 


Jcosh * 


= sinh*. 


(1) 


y = cosh x. 


g = sinh*. 


Jsinh * 


= cosh *. 


(2) 


y tanh x. 


^ = sech 2 *. 
dx 


Jsech 2 * 


= tanh*. 


(3) 


y = coth x. 


-^ = - cosech 2 *. 
dx 


[cosech 2 * 


= - coth *. 


<4) 


y sech x. 


dy sinh * 


Tsinh * 
/ cosh* 


= - sech*. 


(5) 


dx ~ cosh ** 


y - cosech x. 


dy cosh x 


/cosh* 




(6) 


dx sinh * 


sinh * 




y = sinh- 1 *. 


dy 1 


C dx 





(7) 


dx N /*2^n' 


J N/* 2 + 1 






dy 1 


1 dx 




(3) 




dx N ' X 2 _ ! 






y = tanh - l x. 


'= r^* 2 ' X < 1< 


i_dx_ 


= tanh- 1 *. 


(9) 




dy 1 - 


f dx 


coth - l x 


(10) 




dx * 2 - I' ? 


1 * 2 - 1 






^2/ 1 


"f dx 




n ~n 


y sech a *. 


dx *, v /i _ ^2 


J x v /i _ x 




I 11 / 




dii 1 


r /7-7 1 






y = cosech - l x. 




/ 




(12) 


d* * x /* 2 + l 


j* v /* 2 + ] 





EXAMPLES. When integrating algebraic expressions involving the square 
root of a quadratic, hyperbolic functions may frequently be substituted in 



< 114. HVI'KIMlol.K' FUNCTIONS, L'7'.t 

place of the independent variable. Such equations are very common in 
electrotechnics. It is convenient to remember, as in 73, that x = a tanh u, 
= tanh u may be put in place of \/a a - x' 2 , or Vl - x a ; similarly, 
x = a cosh u may be tried in place of \'x 2 - a" 8 ; x = a sinh w for Vx 8 + a 2 . 

(1) Evaluate J x/x 2 a 2 . dx. Substitute x = a sinh M in Vx^+a 2 , and 
x = rt cosh /tin s ./- - <r. 

.-. | v.e^+'rt 2 . fZx- = ia 2 j(cosh 2 w 1) . rf// ; 

= | 2 sinh 2 + ^a 2 = %a sinh zt . a cosh w + ^a 2 !*. 
= & v'i* 2 2 ) * 2 log {x + x /(* 2 a*)}/a + C. 

a 2 x 

The " log " terms can be written -^ sinh ~ a - in the one case, and 

a- '' 

-Q- cosh ~ !- in the other. Verify the next three results : 



Sub8titute x . a 8in h . 




(See page 506.) 

(5) Find the area of the segment OPA (Fig. 107) of the rectangular 
hyperbola x 2 - y 1 = 1. 

Put x = cosh u\ y = sinh w. (See (5), 112.) 

/. Area APM = / y .dx = / sinh 2 z* . dw, 
Ji Jo 

= / (cosh <2u - l).du = % sinh 2 - %u. 

J o 

.-. Area 0PM = $ Area PJtf . OM - Area ^1P3I = %u. 
Note the area. of the circular sector OP' A (same figure) = 0, where 6 is 
the angle AOP'. 

(6) Rectify the catenary curve y = cosh xjc measured from its lowest 
point. Ansr. I = c sinh xjc. Note I = when x = 0, .'. C = 0. 

(7) Rectify the curve y z = lax (see example (1), page 187). The expres- 
sion N /(l + ajx)dx has to be integrated. Hint. Substitute x = a sinh 2 ^. 
2rt Jcosh 2 . du remains. Ansr. = j(l + cosh 2u}du, or a(^ + sinh2w). At 
vertex, where x = 0, sinh u = 0, C 0. 

Show that the portion bounded by an ordinate passing through the 
focus has I = 2-296. Hint. Diagrams are a great help in fixing limits. 
Note x = , .-. sinhw = 1, coshw = v2, from (5), 112. From (1), 113, 
sinh ~ J 1 = u = log(l + \/2). From (20), page 505, sinh 2w = 2 sinh u . cosh M. 

I = |ju + sinh 2it~T = u + sinh u . cosh u = log(l + \'2) + v'2. 
Use Table of Natural Logarithms, Chapter XIII. 

(8) Show that y = A cosh mx + B sinh mx, satisfies the equation of 
d'ty/dx^ = w 2 x, where m, .4 and B are undetermined constants. Note the 
resemblance of this result with a solution of d^y/dx 2 = - n 2 x, which is 
y = A cos nx + B sin nx. 



280 HIGHER MATHEMATICS. 116. 

115. Demoivre's Theorem. 

Refer to the footnote, page 175. Since 

cos #! = i(e t:c i + e - ia; i) ; t sin x l = %(e ix i - e-^), 
and e^i = cos x l + t sin x 1 ; e ~ ta: i = cos o^ t sin x lt 
if we substitute wa; for x lt where n is any real quantity, positive or 
negative, integral or fractional, 

cos nx = ^(e tnx + e ~ tnx ) ; sin nx = \(&- nx - e~ inx ). 
By addition and subtraction and a comparison with the preceding 
expressions, 

cos nx + L sin nx e = (cos x + t sin x) n \ 
cos nx - i sin nx = e - inx = (cos x - L sin x) n ) ' ' 

Note e* = y, (e x ) n = y n , or, e = y n . 
Equations (1) are known as Demoivre's theorem. 

EXAMPLES. (1) Verify the following result and compare it with Demoivre's 

theorem : 

(cos x + i sin #) 2 = (cos 2 x - sin 2 #) + 2t sin x . cos x ; 
cos 2x + t sin 2x. 

(2) Show e* + & = ewP = e*(cos )8 + t sin /3), 

(3) Show j"ez(cos ftx + t sin ftx)dx = eoa;(cos &x + i sin /8a;)/(a + i)8) ; 

(cos &x + i sin flx) (a - t/3) 

a 2 + )8 2 
a j(a cos j8x + )8 sin )8x) + t( - j8 cos 0x + a sin #c) 

a 2 + )8 2 

Demoivre's theorem is employed in algebra in the solving of certain cubic 
equations. The integration of quadratic expressions of the type 

Ax + B 

{(x + a) 2 + fi 2 }*' 

may sometimes be effected by substituting x + a = b tan B ; at others, it is 
recommended to split the quadratic into its so-called conjugate factors, 
x + a + ib, and x + a - tb. Integrate and reduce the result to a real form 
by means of Demoivre's theorem. 

For a fuller discussion on the properties of hyperbolic functions, consult 
Chrystal's Textbook of Algebra, Part ii. (A. & C. Black, London), also Merriman 
and Woodward's Higher Mathematics (Wiley & Sons, New York, 1898), page 
107; and Greenhill's A Cliapter in the Integral Calculus (F. Hodgson, London). 



116. Numerical Values of the Hyperbolic Sines and 

Cosines. 

Tables IV. and V. (pages 510 and 511) contain numerical values 
of the hyperbolic sines and cosines for values of x from to 5, at 



116. HYPERBOLIC FUNCTIONS. 281 

intervals of O'Ol. They have been checked by comparison with 
Des Ingenieurs Taschenbuch, edited by the Hiitte Academy (von 
Ernst & Korn, Berlin, 1877). 

The tables are used exactly like the ordinary logarithmic tables. 

Numerical values of the other functions can be easily deduced 
from those of sinh x and cosh x by the aid of equations (4), 110. 



282 



CHAPTER VII. 
HOW TO SOLVE DIFFERENTIAL EQUATIONS. 

THIS chapter may be looked upon as a sequel to that on the 
integral calculus, but of a more advanced character. The 
" methods of integration " already described will be found ample 
for most physico-chemical processes, but chemists are proving 
every day that more powerful methods will soon have to be 
brought in. As an illustration, I may refer to the set of differ- 
ential equations which Geitel encountered in his study of the 
velocity of hydrolysis of the triglycerides by acetic acid (Journal 
fur praktische Chemie [2], 55, 429, 1897). 

I have previously pointed out that in the effort to find the 
relations between phenomena, the attempt is made to prove that 
if a limited number of hypotheses are prevised, the observed facts 
are a necessary consequence of these assumptions. The modus 
operandi is as follows : 

1. To " anticipate Nature " by means of a " working hy- 
pothesis," which is possibly nothing more than a " convenient 
fiction ". 

"From the practical point of view," says Professor Rucker (Presidential 
Address to the B. A. meeting at Glasgow, September, 1901), " it is a matter of 
secondary importance whether our theories and assumptions are correct, if 
only they guide us to results in accord with facts. . . . By their aid we can 
foresee the results of combinations of causes which would otherwise elude us." 

2. Thence to deduce an equation representing the momentary 
rate of change of the two variables under investigation. 

3. Then to integrate the equation so obtained in order to 
reproduce the "working hypothesis" in a mathematical form 
suitable for experimental verification (see 18, 69, 88, 89, and 
elsewhere). 

So far as we are concerned this is the ultimate object of our 
integration. By the process of integration we are said to solve 
the equation. 



< 117. HOW To SOLYI-: DIFFERENTIAL EQUATIONS 

For the sake of convenience, any equation containing differ- 
entials or differential coefficients will, after this, be called a 
differential equation. 



117. The Solution of a Differential Equation by the 
Separation of the Variables. 

The different equations hitherto considered have required but 
little preliminary arrangement before integration. For example, 
when preparing the equations representing the velocity of a 
chemical reaction of the general type : 

dx/dt = kf(x), .... (1) 

we have invariably collected all the x's to one side, the t's, to the 
other, before proceeding to the integration. 

This separation of the variables is nearly always attempted 
before resorting to other artifices for the solution of the differential 
equation, because the integration is then comparatively simple. 
The following examples will serve to emphasise these remarks : 

EXAMPLES. (1) Integrate the equation, y . dx + x . dy = 0. Rearrange 
the terms so that 



by multiplying through with l/xy. Ansr. log x + log y = C. 

Two or more apparently different answers may be the same. Thus, the 
solution of the preceding equation may also be written, 

logxy log e c , i.e., xy = e; or log xy = log C', i.e., xy = C'. 
C and log C' are, of course, the arbitrary constants of integration. 

(2) The equation for the rectilinear motion of a particle under the in- 
fluence of an attractive force from a fixed point is 

v . dvjdx + nix 2 = 0. 
Solve. Ansr. v 2 = pjx + C. 

(3) Solve (l+x*)dy = ^fy.dx. Ansr. 2 Jy - tan- 1 * = C. 

(4) Solve y - x.dyjdx = a(y + dy/dx). Ansr. y = C(a + x)( l ~ a l 

(5) In consequence of imperfect insulation, the charge on an electrified 
body is dissipated at a rate proportional to the magnitude E of the charge. 
Hence show that if a is a constant depending on the nature of the body, and 
EQ represents the magnitude of the charge when t (time) = 0, 

E = E e-*. . 

Hint. Compound interest law. Integrate by the separation of the variables. 
Interpret your result. 

(6) Abegg's formula for the relation between the dielectric constant (D) of 
a fluid and temperature 6, is 

= D/190. 



284 HIGHER MATHEMATICS. 117. 

Hence show that D = Cte-0/ 190 , where C is a constant whose value is to be 
determined from the conditions of the experiment. Put- the answer into 
words. 

(7) What curves have a slope - x(y to the x-axis ? Ansr. The rectangular 
hyperbolas xy = C. Hint. Set up the proper differential equation and solve. 

(8) The relation between small changes of pressure and volume of a gas 
under adiabatic conditions, is ypdv + vdp = 0. Hence show that pvn constant. 

(9) A lecturer discussing the physical properties of substances at very low 
temperatures, remarked " it appears that the specific heat of a substance de- 
creases with decreasing temperatures at a rate proportional to the specific 
heat of the substance itself ". Set up the differential equation to represent 
this "law" and put your result in a form suitable for experimental verification. 

(10) Helmholtz's equation for the strength of an electric current (C) at the 
time t, is 

E L<W 
L ~ R ~ R dt' 

where E represents the electromotive force in a circuit of resistance R and 
self-induction L. If E, R, L, are constants, show that RC = E(l - e~ RtlL ) 
provided C = 0, when t = 0. 

A substitution will often enable an equation to be treated by 
this simple method of solution. 

EXAMPLE. Solve (x - y*)dx + 2xydy = 0. Ansr. xe^ lx = C. Hint, put 
t/ 2 = v, divide by x 2 , .. dx/x + d(vjx) = 0, etc. 

If the equation is homogeneous in x and y, that is to say, if 
the sum of the exponents of the variables in each term is of the 
same degree, a preliminary substitution of x = ty, or y = to, ac- 
cording to convenience, will always enable variables to be separated. 
The rule for the substitution is to treat the differential coefficient 
which involves the smallest number of terms. 

EXAMPLES. (1) Solve x + y . dy/dx - 2y = 0. Substitute y = tx, 



Ansr. (x - y)e*H*-) = C. 

(2) If (y - x)dy + ydx = 0, y = Ce~*b. 

(3) If x*dy - y*dx - xydx = 0, x = *'+ C. 

(4) (x 2 + if)dx = Zxydy, x* - y* = Cx. 

Non-homogeneous equations in x and y can be converted into 
the homogeneous form by a suitable substitution. 

The most general type of a non-homogeneous equation of the 
first degree is, 

(ax + by + c)dx + (a'x + b'y + c')dy = 0. . (2) 



< 117. HOW TO SOLVE DIFFERENTIAL EQl'A'l l<>\- JM.% 

To convert this into an homogeneous equation, assume that 

x = v + h and y = w + k, 
and substitute in the given equation (2). Thus, we obtain 

\av + bw + (ah + bk + c)}dv +\a'v + b'w + (a'h + b'k + c')\dw = 0. (3) 
Find h and k so that 

ah + bk + c = ; a'h + b'k + c' = 0. 
, b'c - be' ac' - a'c 

.-. h = _~ -- - ; k = -77 -- jj. . . (4) 

a b - ab ' $b - ab 

Substitute these values of h and k in (3). The resulting equation 
(av + biv)dv + (a'v + b'w)dw = 0, . (5) 

is homogeneous and, therefore, may be solved as just indicated. 

EXAMPLES. (1) Solve (3y - 7x - l)dx + (ly - 3x - 3)dy = 0. Ansr. 
(y - x - 1)% + x + I) 6 = C. Hints. From (2), a = - 7, 6 = 3, c = -7; 
a' = - 3, b' = 7, c' = - 3. From (4), h = - 1, k = 0. Hence, from (3), 

Sivdv - Ivdv + Itcdw - Svdw = 0. 

To solve this homogeneous equation, substitute w = vt, as above, and separate 
the variables. 

dv 3 - It -,. fdv 

' 



.-. 7 log v + 2 log(* - 1) + 5 log(t + 1) = C ; or, v>(t - l) 2 (i + 1)' = C. 

But x = v + h, .. v = x + 1 ; y = w + k, .-. y = w ; .. t = wfv = yf(x + 1), etc. 

(2) If (2y - x - l)dy + (2z - y + l)dx = 0, x 2 - xy + i/ 2 + x - y = C. 

If in (3), 

a : b = a' : b' = 1 : m (say), 
h and k are indeterminate, since (2) then becomes, 

(ax + by + c)dx + {m(ax + by) + c'}dy = 0. 
The denominators in equations (4) also vanish. In this case put 

z = ax + by 
and eliminate y, thus, we obtain, 

, z + c dz n 

a + b _ + = 0, . . . (6) 
mz + c ax 

an equation which allows the variables to be separated. 

EXAMPLES. (1) Solve (2x + Sy - 5)dy + (2x + Sy - l)dx = 0. 

Ansr. x + y - 4 log(2x- + Sy + 7) = C. 
(2) Solve (3y + 2x + )dx - (x + &y + 5)dy = 0. 

Ansr. 91og{(2l7/ + Ux + 22)/7} - 21(2# - x) = C. 



When the variables cannot be separated in a satisfactory manner, 
special artifices must be adopted. We shall find it the simplest 
plan to adopt the routine method of referring each artifice to the 
particular class of equation which it is best calculated to solve. 



286 HIGHER MATHEMATICS. $ 118. 

These special devices are sometimes far neater and quicker pro- 
cesses of solution than the method just described. 

We shall follow the conventional x and y rather more closely 
than in the earlier part of this work. The reader will know, by 
this time, that his x and y's, his p and i?'s and his s and t's are 
not to be kept in " water-tight compartments ", 

It is perhaps necessary to make a few general remarks on the 
nomenclature. 



118. What is a Differential Equation? 

We have seen that the straight line, 

y = mx + b, . . . . (1) 
fulfils two special conditions : 

1. It cuts one of the coordinate axes at a distance b from the 
-origin. 

2. It makes an angle tan a = m, with the #-axis. 
By differentiation. 

? = (2) 

ax 

This equation has nothing at all to say about the constant b. 
That condition has been eliminated. Equation (2), therefore, 
represents a straight line fulfilling one condition, namely, that 
it makes an angle tan -1 w with the ic-axis. 

Now substitute (2) in (1), the resulting equation, 

-& + * . . . - (3) 

in virtue of the constant b, satisfies only one definite condition, 
(3), therefore, is the equation of any straight line passing through 
b. Nothing is said about the magnitude of the angle tan ~ l m. 
Differentiate (2). The resulting equation, 

^-0 (4) 

dx 2 

represents any straight line whatever. The special conditions 
imposed by the constants m and b in (1), have been entirely 
eliminated. Equation (4) is the most general equation of a 
straight line possible, for it may be applied to any straight line 
that can be drawn in a plane. 

Let us now find a physical meaning for the differential equa- 
tion. 



us. HOW TO SOLVK mn-'KKKN i IAL K^TATION- 



In $ 7, we have found that the third differential coefficient, 

// 3 represents "the rate of change of acceleration from moment 
to moment ". Suppose that the acceleration d' 2 s/dt' 2 , of a moving 
body does not change or vary in any way. It is apparent that the 
i ;itc of change of a constant or uniform acceleration must be zero. 
In mathematical language, this is written, 

d*s/dt* = 0. .' . . . (0) 
Now integrate this equation once. We obtain, 

d 2 s/dt 2 = constant, say = g. . . (6) 

Equation (6) tells us not only that the acceleration is constant, but 
it fixes that value to the definite magnitude <j ft. per second. 

But acceleration measures the rate of change of velocity. In- 
tegrate (6), we get, 

dsfdt = gt + C l ..... (7) 

From 72, we have learnt how to find the meaning of C v Put 
t = 0, then dx/dt = C r This means that when we begin to reckon 
the velocity, the body may have been moving with a definite velocity 
(\ . Let Cj = v ft. per second. Of course, if the body started from 
a position of rest, C l = 0. 

Now integrate (7) and find the value of C 2 in the result, 

s = 1^2 + V( f + C 2 , , .".... (8) 

by putting t = 0. It is thus apparent that C. 2 represents the space 
which the body had traversed when we began to study its motion. 
Let C. 2 = s ft. The resulting equation 

^ = \gP + V + * 0> . . (9) 

tells us three different things about the moving body at the instant 
we began to take its motion into consideration. 

1. It had traversed a distance of s ft. To use a sporting 
phrase, if the body is starting from "scratch," s = 0. 

2. The body was moving with a velocity of i' ft. per second. 

3. The velocity was increasing at the uniform rate of #ft. per 
second. 

Equation (7) tells us the two latter facts about the moving 
body ; equation (6) only tells us the third fact ; equation (5) tells 
us nothing more than that the acceleration is constant. (5), there- 
fore, is true of the motion of any body moving with a uniform 
acceleration. 

EXAMPLE. If a body falls in the air, experiment shows that the retarding 
effect of the resisting air is proportional to the square of the velocity of the 
moving body. Instead of g, therefore, we must write g - /30 s , where is the 



288 HIGHER MATHEMATICS. 118. 

variation constant of page 487. For the sake of simplicity, put ft = g/a? and" 
show that 

gfftla _ e - gtla ft 2 e gtla + c -gtla a 2 Q t 

v - V + .-.. ; S - g 10g - -V - - (,- J g C 8h i 
since w = when i = 0, and s = when < = 0. 

Similar reasoning holds good from whatever sources we may 
draw our illustrations. We are, therefore, able to say that a, 
differential equation, freed from constants, is the most general way 
of expressing a natural law. 

Any equation can be freed from its constants by combining it 
with the various equations obtained by differentiation of the given 
equation as many times as there are constants. The operation is 
called elimination. 

EXAMPLES. (1) Eliminate the arbitrary constants a and 6, from 

y = ax + bx z . 

Differentiate twice and combine the results with the original equation. The 
result, 



is quite free from the arbitrary restrictions imposed in virtue of the presence 
of the constants a and b in the original equation. 

(2) Eliminate m from y 2 = 4=mx. Ansr. y 2 = 2x . dy/dx. 

(3) Eliminate a and ft from y a cos x + sin x. Ansr. d z y/dx 2 + y = 0. 

(4) Eliminate a and ft from y = ae ax + fte bx . 

Ansr. d 2 y/dx* - (a + b) . dyfdx + aby = 0. 

(5) Eliminate k from dx/dt = k(a - x) of 69. What does the resulting. 
equation mean ? 

We always assume that every differential equation has been 
obtained by the elimination of constants from a given equation 
called the primitive. In practical work we are not so much 
concerned with the building up of a differential equation by the 
elimination of constants from the primitive, as with the reverse 
operation of finding the primitive from which the differential 
equation has been derived. In other words, we have to find 
some relation between the variables which will satisfy the differ- 
ential equation. Given an expression involving x, y, dx/dy,, 
d 2 x/dy 2 , . . ., to find an equation containing only x, y and 
constants which can be reconverted into the original equation 
by the elimination of the constants. 

This relation between the variables and constants which satisfies 
the given differential equation is called a general solution, or a 
complete solution, or a complete integral of the differential 



$ ll'.. HOW TO SOLVE DIFFERENTIAL EQUATION- _'*'. 

equation. A solution obtained by giving particular values to the 
arbitrary constants of the complete solution is a particular solution. 

Thus y = mx is a complete solution oiy = x. dy/dx ; y = x tan 45, 
is a particular solution. 

A differential equation is ordinary or partial, according as 
there is one or more than one independent variables present. 
Ordinary differential equations will be treated first. 

Equations like (2) and (3) above, are said to be of the first 
order, because the highest derivative present is of the first order. 
For a similar reason (4) and (6) are of the second order, (5) of the 
third order. The order of a differential equation, therefore, is 
fixed by that of the highest differential coefficient it contains. The 
degree of a differential equation is the highest power of the 
highest order of differential coefficient it contains. Thus, 

g + tfft* +**: A 

dx 2 \dx) 
is of the second order and third degree. 

It is not difficult to show that the complete integral of a differ- 
ential equation of the nth order, contains n and only n arbitrary 
constants. 

We shall first consider equations of the first order. 

119. Exact Differential Equations of the First Order. 

The reason many differential equations are so difficult to solve 
is due to the fact that they have been formed by the elimination 
of constants as well as by the elision of some common factor from 
the primitive. Such an equation, therefore, does not actually re- 
present the complete or total differential of the original equation 
or primitive. The equation is then said to be inexact. On the 
other hand, an exact differential equation is one that has been 
obtained by the differentiation of a function of x and y and per- 
forming no other operation involving x and y. 

Easy tests were described in 24, 25, to determine whether 
any given differential equation is exact or inexact. It was pointed 
out that the differential equation, 

H.dx + N.dy = Q, . . . (1) 

is the direct result of the differentiation of any function , provided, 



290 HIGHER MATHEMATICS. 119. 

This last result was called the criterion of integrability, because, 
if an equation satisfies the test, the integration can be readily 
performed by a direct process. This is not meant to imply that 
only such equations can be integrated as satisfy the test, for many 
equations which do not satisfy the test can be solved in other 
ways. 

EXAMPLES. (1) Apply the test to the equations, 

ydx + xdy = 0, and ydx - xdy = 0. 
In the former, M = y, N x ; 

.-. VM/dy = 1, -dNftx = I ; .-. dM/Vy = dN/dx. 

The test is, therefore, satisfied and the equation is exact. In the other 
equation, M = y, N = - x, 

.-. dM/dy = 1, 'dNl'dx = - l. 

This does not satisfy the test. In consequence, the equation cannot be solved 
by the method for exact differential equations. 

(2) Is the equation, (x + 2y)xdx + (x 2 - y*)dy = 0, exact ? M = x(x + 2y), 
N = z 2 - if ; .-. *dM[dy = 2x, 'dNl'dx = 2x. The condition is satisfied, the 
.equation is exact. 

(3) Show that (a?y + x*)dx + (6 3 + a?x)dy = 0, is exact. 

^4) Show that (sin y + y cos x)dx + (sin x + x cos y)dy = 0, is exact. 

To integrate an equation which satisfies the criterion of in- 
tegrability, we must remember that M is the differential coefficient 
of u with respect to x, y being constant, and N is the differential 
coefficient of u with respect to y, x being constant. Hence we 
may integrate Mdx on the supposition that y is constant and then 
treat Ndy as if x were a constant. The complete solution of the 
whole equation is obtained by equating the* sum of these two 
integrals to an undetermined constant. The complete integral is 

u = C (3) 

EXAMPLES. (1) Integrate x(x + 2y)dx + (x 2 - y 2 )dy = 0, from the pre- 
ceding set of examples. Since the equation is exact, 
M=x(x + 2y) ; JV = .r 2 - y 2 ; 
.-. jMdx = jx(x + 2y)dx = %x* + x*y = Y, 

where Yis the integration constant which may, or may not, contain y, because 
y has here been regarded as a constant. 
Now the result of differentiating 

$x* + x*y = Y, 
should be the original equation. On trial, , 

x 2 dx + Zxydx + x*dy = dY. 
On comparison with the original equation, it is apparent that 

dY=y*dy; .'. Y = $y* + C. 

Substitute this in the preceding result. The complete solution is, therefore, 
la-s + x* - i* = C. 



.< ll'.. HO\V T<> SOI.VK nil-TKKKNTIAL \-'.<}\ A IK >\ - 291 

To summarise: The method detailed in the example just given may be 
put into a more practical shape. 

To integrate an exact differential equation, first find JA/ .dr on the as- 
sumption that y is constant and substitute the result in 



C. ... (4) 

With the old example, therefore, having found \Mdx, we may write down at 
once 



.-. Ix* + x*y + J(z 2 - 7/2 _ x z) dy = C . 
And the old result follows directly. If we had wished we could have used 

I Ndy + f(M - J^fNdy\dx = C, 

in place of (4). 

In practice it is often convenient to modify this procedure. If the equa- 
tion satisfies the criterion of integrability, we can easily pick out terms which 
make Mdx + Ndy = 0, and get 

Mdx + Y and Ndy + X, 

where Y cannot contain x and X cannot contain ?/. Hence if we find Mdy 
and Ndx, the functions X and F will be determined. 

In the above equation, the only terms containing x and y are Ixydx + x 2 dy t 
which obviously have been derived from .r 2 ?/. Hence integration of these and 
the omitted terms gives the above result. 

(2) Solve (x 2 - 4xy - 2y 2 )dx + (?/ 2 - xy - 2x 2 )dy = 0. Pick out terms 
in .1- and y, we get 

- (xy + 2if)dx - (4xy + 2x*)dy = 0. 
Integrate. .. - 2x 2 y - ~2xy 2 = constant. 

Pick out the omitted terms and integrate for the complete solution. We get, 
jx 2 dx + jy 2 dy - 2x 2 y - 2xy 2 = x 3 - Gx 2 ?/ - 6xy* + if = constant. 

(3) Show that the .solution of (a?y + x?)dx + (6 :{ + a?x)dy = 0, is 

a?xy + b s y + $s? = C. Use (4). 

(4) Solve (x 2 - y*)dx - Zxydy = 0. Ansr. ^x 2 - y 2 = Cfx. Use (4). 

Equations made exact by means of integrating factors. As just 
pointed out, the reason any differential equation does not satisfy the 
criterion of exactness, is because the " integrating factor " has been 
cancelled out during the genesis of the equation from its primitive. 

If, therefore, the equation 

Mdx + Ndy = 0, 

does not satisfy the criterion of integrability, it will do so when 
the factor, previously divided out, is restored. Thus, the pre- 
ceding equation is made exact by multiplying through with the 
integrating factor /x. Hence, 

p.(Mdx + Ndy) = 0, 

satisfies the criterion of exactness, and the solution can be obtained 
as described above. 



292 HIGHER MATHEMATICS. 120. 

120. How to find Integrating Factors. 

Sometimes integrating factors are so. simple that they can be 
detected by simple inspection. 

EXAMPLES. (1) ydx - xdy = is inexact. It becomes exact by multipli- 
cation with either x~ 2 , x~ l . y 1 , or y 2 . 

(2) In (y - x)dy + ydx = 0, the term containing ydx - xdy is not exact, 
but becomes so when multiplied as in the preceding example. 

... *L - x ^ - y dx = ; or log y - * = C. 

y y 2 y 

For the general theorems concerning the properties of integrating factors, 
the reader must consult some special treatise, say Boole's A Treatise on 
Differential Equations, pages 55 et seq., 1865. 

We have already established, in 26, that an integrating factor 
always exists which will make the equation 

Mdx + Ndy = 0, 
an exact differential. 

Moreover, there is also an infinite number of such factors, for 
if the equation is made exact when multiplied by ju,, it will remain 
exact when multiplied by any function of /x. 

The different integrating factors correspond to the various forms 
in which the solution of the equation may present itself. For 
instance, the integrating factor x~ l y~ l , of ydx + xdy = 0, corre- 
sponds with the solution log x + log y = C. The factor y ~ 2 corre- 
sponds with the solution xy C". 

Unfortunately, it is of no assistance to know that every 
differential equation has an infinite number of integrating factors. 
No general practical method is known for finding them. Here 
are a few elementary rules applicable to special cases. 

Rule I. Since 

d(x m y") = x m ~ 1 y n ~ l (mydx + nxdy), 

an expression of the type mydx + nxdy = 0, has an integrating 
factor x m ~ 1 y n ~ 1 ', or, the expression 

x a y B (mydx + nxdy) = 0, . . (1) 

has an integrating factor 



or more generally still, 

^-i-Y-i-0, . (2) 

where k may have any value whatever. 

EXAMPLE. Find an integrating factor of ydx - xdy = 0. Here, o = 0, 
/8 = 0, m = 1, n = - 1 .. ?/~ 2 is an integrating factor of the given equation. 



j IL'O. HOW TO SOLVE DIFFKUKM I Al. K^TATIONS. -J'.:; 

If the expression can be written 

x?yt(mydx '+ nxdy) + x*'yP(m'ydx + n'xdy) = 0, . (3) 
the integrating factor can be readily obtained, for 

z*'-i -<y- 1-0; and :**'""- ^y""- 1 -*', 

are integrating factors of the first and second members respectively. 
In order that these factors may be identical, 

km - 1 - a = k'm' - I - a ; kn - 1 - ft = k'ri - 1 - ft'. 
Values of k and k' can be obtained to satisfy these two conditions 
by solving these two equations. Thus, 

k = n'(a - a) - m'(ft - ft') k , = n(a - a) - m(ft - ft') 
mri - mn mri - m'n 

EXAMPLES. (1) Solve y*(ydx - 2xdy) + x 4 (2ydx + xdy) = 0. Hints. Show 
that a = 0, 3 = 3, m = 1, n = - 2 ; a' = 4, j8' = 0, m' = 2, ri = 1 ; .-. 
x k-iy-2k-4 j s an integrating factor of the first, a 2 *'- 5 !/*'- 1 of the second 
^member. Hence, from (4), k = - 2, k' = 1, .-. , x~ 3 is an integrating factor 
of the whole expression. Multiply through and integrate for 2x*y - y* = Cx 2 - 

(2) Solve (y 3 - 2yx z )dx + (2xy 2 - x 3 )dy = 0. Ansr. x z y*(y* - x 2 ) = C. In- 
tegrating factor deduced after rearranging the equation is xy. 

Rule II. If the equation is homogeneous and of the form : 
Mdx + Ndy = 0, then (Mx + Ny) ~ 1 is an integrating factor. 
Let the expression 

Mdx + Ndy = 0, 

be of the mth degree and /x an integrating factor of the ?*th degree, 
.-. pMdx + fjiNdy = du, ... (5) 

is of the (w + ?i)th degree, and the integral n is of the (w + w + l)th 
degree. 

By Euler's theorem, 22, 

.-. fjiMx + fj.Ny = (m + n + l)u. . . (6) 
Divide (5) by (6), 

Mdx + Ndy _ 1 du 

MX + Ny m + n + 1 u ' 

The right side of this equation is a complete differential, conse- 
quently, the left side is also a complete differential. Therefore, 
(Mx -|- Ny)~ l has made Mdx + Ndy = an exact differential 
equation. 

EXAMPLES. (1) Show that (x*y - xy*)- 1 is an integrating factor of 
(z 2 ?/ + y*)dx - 2xy-dy = 0. 

(2) Show that ll(x' 2 -ny + if) is an integrating factor of ydy + (j:-ny)dx = Q. 

The method, of course, cannot be used if Mx + Ny is equal to zero. In 
this case, we may write y = Ox, a solution. 



294 HIGHER MATHEMATICS. 120. 

Rule III. If the equation is of the form, 

fi(, y)ydx + fz(x, y)xdy = 0, 
then (Mx Ny)~ l is an integrating factor. 

EXAMPLE. Solve (1 + xy)ydx + (1 - xy)xdy 0. Hint. Show that the 
integrating factor is l/2x 2 y 2 . Divide out . .. $Mdx = \\xy + \ogx. Ansr. 
x = Cye~ llx y. 

If MX - Ny = 0, the method fails and xy = C is then a solution of the 
equation. E.g., (1 + xy)ydx + (1 + xy)xdy = 0. 

Rule IY. If if - \ is a function of x only, e f dx is an 
N\ oy ox ) 

integrating factor. Or, if W ^ --- -y- J = f(y), then e rf(y)dy is an 
integrating factor. These are important results. 

EXAMPLES. (1) Solve (x 2 + y^dx - Zxydy = 0. Ansr. x 2 - y 2 = Cx. 
Hint. Show f(x) = - 2/<c. The integrating factor is, therefore, 

e ~f2dx]x _ e log I/a: 2 _ lj x 2 f 

(Why ?) Prove that this is an integrating factor, and solve as in the pre- 
ceding section. 

(2) Solve (7/ 4 + 2y)dx + (xy 3 + 2y 4 -4:x)dy = 0. Ansr. xy* + y 4 + 2x = Cy 2 . 

(3) We may prove the rule for a special case in the following manner. 
The steps will serve to recall some of the principles established in some 
earlier chapters. 

Let *L + Py = Q, ..... (Q 

where P and Q are either constant or functions of x. Let p be an integrating 
factor which makes 

dy + (Py - Q)dx = 0, .... (8) 

an exact differential. 

.'. pdy + n(Py - Q)dx = Ndy + Mdx. 



I - 



= (Py - Q&dx + 
dy 

= - dy + Padx. 

+ ]^ 
dy 



and since \og,.e = 1. 

($Pdx)loge = logjw; .'. yu. = e-' rdx . ... (9) 
This result will be employed in dealing with linear equations, 122. 



;j lui. MOW To >OLYK DIFFERENT] \i. !:< >i \ I IONS. -".:, 

121. The First Law of Thermodynamics. 

According to the discussion at the end of the first chapter, one 
way of stating the first law of thermodynamics is as follows : 

dQ = dU + dW, 

which means that when a quantity of heat, dQ, is added to a 
substance, one part of the heat is spent in changing the internal 
energy, dU, of the substance and another part, dW, is spent in 
doing work against external forces. In the special case, when 
that work is expansion against atmospheric pressure, dW = p . dv, 
as shown in 91. See (11), page 524. 

We know that the condition of a substance is completely 
defined by any two of the three variables p, v, 0, because when 
any two of these three variables is known, the third can be deduced 
from the relation 

pv = BO. 

Hence it is assumed that the internal energy of the substance is 
completely defined when any two of these variables are known. 

Now let the substance pass from any state A to another state 
B (Fig. 111). The internal energy of the substance in the state B 
is completely determined by the coordin- 
f ates of that point, because U is quite 
independent of the nature of the trans- 
formation from the state A to the state B. 
It makes no difference to the magnitude of 
^ U whether that path has been via APB 
or AQB. In this case U is said to be a 
FIG. ill. single-valued function completely defined 

by the coordinates of the point corresponding to any given state. 
In other words, dU is a complete differential. Hence 



is an exact differential equation, where x and y represent any pair 
of the variables p, r, 0. 

On the other hand, the external work done during the trans- 
formation from the one state to another, depends not only on the 
initial and final states of the substance, but also on the nature of 
the path described in passing from the state A to the state B. 
For example, the substance may perform the work represented by 
the area AQBB'A' or by the area APBB'A', in its passage from the 




296 HIGHER MATHEMATICS. 122. 

state A to the state B. In fact the total work done in the passage 
from A to B and back again, is represented by the area APBQ 
(page 183). In order to know the work done during the passage 
from the state A to the state B, it is not only necessary to know 
the initial and final states of the substance as denned by the co- 
ordinates of the points A and B, but we must know the nature of 
the path from the one state to the other. 

Similarly, the quantity of heat supplied to the body in passing 
from one state to the other, not only depends on the initial and 
final states of the substance but also on the nature of the trans- 
formation. 

All this is implied when it is said that " dW and dQ are not 
perfect differentials ". Although we can write 



we must put, in the case of W or Q, 



Therefore the partial differentiation of x with respect to y, furnishes 
a complete differential equation only when we multiply through 
with the integrating factor /*, so that 



where x and y may represent any pair of the variables p, v, 0. 

The integrating factor is proved in thermodynamics to be equiva- 
lent to the so-called Carnot's function (see Preston's Theory of Heat). 

To indicate that dPFand dQ are not perfect differentials, some 
writers superscribe a comma to the top right-hand corner of the 
differential sign. The above equation would then be written, 
d'Q = dU + d'W. 



122. Linear Differential Equations of the First Order. 

A linear differential equation of the first order involves only 
the first power of the dependent variable y and of its first 
differential coefficients. The general type is, 

s + ^-e 

where P and Q may be functions of x, or constants. 



122. HOW TO SOLVE WKFKUKNTIAI. EQUATIONS 297 

We have just proved that ef pd * is an integrating factor of (1), 
therefore 

e fpdx (dy + Pydx) = e jrdx Qdx, 
is an exact differential equation. The general solution is, 

ye /N* = le**Qdx + C. . . . (2) 

The linear equation is one of the most important in applied mathe- 
matics. In particular cases the integrating factor may assume a 
very simple form. 

In the following examples, remember that e log * = x, .. if 
= logrc, e' l>djc = x. 



EXAMPLES. (1) Solve (1 + x*)dy = (m + xy}dx. Reduce to the form (1) 
and we obtain 

dy_ _ x _ m 
dx 1 + x* y ~ I + x 2 ' 



\Pdx = - ^^ = - \ log (1 + * 2 ) - - log 



Remembering log 1 = 0, loge = 1, the integrating factor is evidently, 

log e rpox = log 1 _ log \/l + a; 2 , or ef p ** = ^ + ^. 

Multiply the original equation with this integrating factor, and solve the 
resulting exact equation as 119, (4), or, better still, by (2) above. The 
solution : y = mx + C *J(1 + x 2 ) follows at once. 

(2) Ohm's law far a variable current flowing in a circuit with a coefficient 
of self-induction L (henries), a resistance R (ohms), and a current of C 
((amperes) and an electromotive force E (volts), is given by the equation, 

E = RC + lg. 

This equation has the standard linear form (1). If E is constant, show that 
the solution is, 

C = E/R + Be- RtlL , 

where B is the arbitrary constant of integration (page 159). Show that C 
Approximates to E/R after the current has been flowing some time (/). Hint 
for solution. Integrating factor is e l{tlL . 

(3) The equation of motion of a particle subject to a resistance varying 
directly as the velocity and as some force which is a given function of the 
time, is 

dv/dt + AT = /(/). 

Show that v = Ce- kt + e ~ kt }'e kt f(t)dt. 

If the force is gravitational, say g, 

v = Ce - kt + g\k. 

(4) Solve xdy + ydx = x?dx. Integrating factor = x. Ansr. y = [r 3 + C/r. 

Many equations may be transformed into the linear type of 
equation, by a change in the variable. Thus, in the so-called 
Bernoulli's equation, 

dyldx + Py = Qy" ..... (3) 



298 HIGHER MATHEMATICS. 123. 

Divide by y", multiply by (1 - n) and substitute y l ~ n = v, in the 
result. Thus, 



and dv/dx + (1 - n)Pv = Q(l - TO), 

which is linear in v. Hence, the solution is 

C. 
C. 



EXAMPLES. (1) Solve cfy/dz + T//X = y' 2 . Treat as above, substituting 
v = \\y. The integration factor is e -/<**/* =e- lo * x = I/a. 
Ansr. CXT/ - a, 1 // log a; = 1. 

(2) Solve dy/da + a; sin 2 ?/ = jj :J cos 2 ?/. Divide by cos 2 ?/. Put tan # = v.. 
The integration factor is e/ 2 ***, i^ M ^ 2 . Ansr. e* 2 tan y - Je^(a; 2 - 1) = C. 
Hint to solve ve* 2 = jx s e x2 dx + C. Put a; 2 = ,.. 2a%fo = dz, and this integral 
becomes ^ze z dz, or -ie z (^ - 1), etc. 

(3) Here is an instructive differential equation, which Harcourt and Esson- 
encountered during their work on chemical dynamics in '66. 

I.4 + 5_^.o 

y' 2 dx y x 

I shall give a method of solution in full, so as to revise some preceding work. 
The equation has the same form as Bernoulli's. Therefore, substitute 
1 . dv 1 dy 



.. 

^a; x 

an equation linear in v. The integrating factor is 

e- fp **, or, e -**; g, in (2), - - j 

therefore, from (2) ve ~ Kx = - / e ~ Kjr dx + C. 

From 108, 



x 1.2 1.2.3 

But v = l/y. Multiply through with ye x *, and integrate. 



We shall require this result on page 333. 

123. Differential Equations of the First Order and of the 
First or Higher Degree. Solution by Differentiation. 

Case i. The equation can be split up into factors. If the 
differential equation can be resolved into n factors of the first 
degree, equate each factor to zero and solve each of the n equa- 



jj 133, IIO\V To sol.YK IHl-TKKKNTI \l. EQUATIONS, _".. 

tions separately. The n solutions may be left either distinct, or 
combined into one. 

EXAMPLES. (1) Solve x(dxldy)* = y. Resolve into factors of the first 
degree, 

dx/dy = ^'yjx. 
Separate the variables and integrate, 

x /x ^'y= N /C, 
which, on rationalisation, becomes 

(* - y) z - 2C(x + y) + C 2 = 0. 

Geometrically this equation represents a system of parabolic curves each 
of which touches the axis at a distance C from the origin. The separate 
equations of the above solution merely represent different branches of the 
same parabola. 

(2) Solve .i-y(dyldx)~ - (x- - y 2 )dy/dx - xy = 0. Ansr. xy = C, or x 2 - y 2 = C. 
Hint. Factors (,rp + y) (yp - x), where p = dyjdx. 

(3) Solve (dyldxY - Idyjdx + 12 = 0. Ansr. y = 4x + C, or 3* + C. 

Case ii. The equation cannot be resolved into factors, but it can 
be solved for x, y, dy/dx, or y/x. An equation which cannot be 
resolved into factors, can often be expressed in terms of x, y, dy/dx, 
or y x, according to circumstances. The differential coefficient of 
the one variable with respect to the other may be then obtained 
by solving for dyi'dx and using the result to eliminate dy/dx from 
the given equation. 

EXAMPLES. (1) Solve dyjdx + 2xy = x* + if. Since (x - y)- = x 2 - 2xy + y\ 
y = x + \dyldx. 

Diflerentiate ***-' 1 + S 

Separate the variables x and p, where p = dyjdx, and solve for dyjdx, 

Idjj C + ** 



.-. Ansr. y = x + (C + e**)/(C - ?**). 

(2) Solve x(dyldx) z - toj(dyjdx) + ax = 0. Ansr. y = $C(x* + o/C). Hint. 
Substitute for p. Solve for y and differentiate. Substitute pdx for dy, and 
clear of fractions. The variables p and x can be separated. Integrate. 
p = xC. Substitute in the given equation for the answer. 

(3) Solve y(dyldx)* + 2x(dyldx) - y > = 0. Ansr. y* - C(2x + C). Hint. 
Solve for x. Differentiate and substitute dy/p for dx, and proceed as in 
example (2). yp = C, etc. 

Case iii. The equation cannot be resolved into factors, x or y is 
absent. If x is absent solve for dy/dx or y according to conveni- 
ence ; if y is absent, solve for dx/dy or x. Differentiate the result 
with respect to the absent letter if necessary and solve in the 
regular way. 



300 HIGHER MATHEMATICS. 124. 

EXAMPLES. (1) Solve (dyjdx) 2 + x(dy/dx) + 1 = 0. For the sake of greater 
ease, substitute > for dx/dy. The given equation thus reduces to 

x=p + l/p (1) 

Differentiate with regard to the absent letter y, thus, 

p = (l- llp*)dpldy; or, dy/dp = 1/p - l/p". 

.-. y = log_p + l/2p 2 + C (2) 

Combining (1) and (2), we get the required solution. 

(2) Solve dy/dx = y + 1/y. Ansr. 7/ 2 = Ce 2 * - 1. 

(3) Solve dyjdx = x + I/or. Ansr. y = z 2 + log x + C. 



124. Glairaut's Equation. 

The general type of this equation is, 



or, writing p = dy/dx, for the sake of convenience, 

y=px+f(p). . (2) 

Many equations of the first degree in x and y can be reduced 
to this form by a more or less obvious transformation of the vari- 
ables, and solved in the following way : 

Differentiate (2) with respect to x, and equate the result to zero. 



Hence either = ; or, x + f(p) = 0. 

d x 

If the former, 

dp/dx = 0; .-.p = C, 
where C is an arbitrary constant. Hence, 

dy = Cdx ; or, y = Cx + /(C), 
is a solution of the given equation. 

Again, p in x + f(p) may be a solution of the given equation. 
To find p, eliminate p between 

y = px + f(p), and x + f(p) = 0. 

The resulting equation between x and y also satisfies the given 
equation. 

There are thus two classes of solutions to Glairaut's equation. 

EXAMPLES. Find both solutions in the following equations : 

(1) y = px + p 2 . Ansr. Cx + C 2 = y and x 2 + ty = 0. 

(2) (y - px} (p - 1) = p. Ansr. (y - Cx} (C - 1) = C ; N /2/ + v /.r = 1. Bead 
over 67. 



125. HOW TO SOLYK 1>I KFKUKNTIAI. EQUATIONS. '."I 

125. Singular Solutions. 

Clairaut's equation introduces us to a new idea. Hitherto we 
have assumed that whenever a function of x and y satisfies an 
equation, that function plus an arbitrary constant, represents the 
complete or general solution. We now find that functions of x 
and y can sometimes be found to satisfy the given equation, which, 
unlike the particular solution, are not included in the general 
solution. 

This function must be considered a solution, because it satisfies 
the given equation. But the existence of such a solution is quite 
an accidental property confined to special equations, hence their 
cognomen, singular solutions. 

To take the simple illustration of page 142, 

y = px + alp. . (1) 

Eemembering that p has been written for dy/dx, differentiate with 
respect to x, we get, on rearranging terms, 
(x - a/p 2 )dp/dx = 0, 
where either x - a/p 2 = ; dp/dx = 0. 

If the latter, 

p = C ; or, y = Cx + a/C. (2) 

If the former, p = Ja/x, which gives, when substituted in (l) r 
the solution, 

y 2 = ax (3) 

This solution is not included in the general solution, but yet it 
satisfies the given equation. (3) is the singular solution of (1). 

Equation* (2), the complete solution of (1), has been shown to 
represent a system of straight lines which differ only in the value 
of the arbitrary constant C ; equation (3), containing no arbitrary 
constant, is an equation to the common parabola. A point moving 
on this parabola has, at any instant, the same value of dy/dx as if 
it were moving on the tangent of the parabola, or on one of the 
straight lines of equation (2). The singular solution of a differential 
equation is geometrically equivalent to the envelope of the family of 
curves represented by the general solution. The singular solution 
is distinguished from the particular solution, in that the latter is 
contained in the general solution, the former is not. 

Again referring to Fig. 78, it will be noticed that for any point 
on the envelope, there are two equal values of p or dy/dx, one for 
the parabola, one for the straight line. 



302 



HIGHER MATHEMATICS. 



125. 



In order that the quadratic 

ax 2 + bx + c = 0, 
may have equal roots, it is necessary (page 388) that 

b 2 = 4ac ; or, b 2 - 4ac = 0. 
This relation is called the discriminant. From (1), since 

y = px + a/p ; .*, xp 2 - yp + a = 0. 
In order that equation (5) may have equal roots, 



(5) 



as in (4). This relation is the locus of all points for which two values 
of p become equal, hence it is called the p-discriminant of (1). 

In the same way if C be regarded as variable in the general 
solution (2), 

y = Cx + a/C ; or, xC 2 - yC + a = 0. 
The condition for equal roots, is that 

y 2 = &ax, 

which is the locus of all points for which the value of C is the 
same. It is called t^e C-discriminant. 

Before applying these ideas to special cases, we may note that 
the envelope locus may be a single curve (Fig. 78) or several 
(Fig. 79). For an exhaustive discussion of the properties of 
these discriminant relations I must refer the reader to the 
numerous textbooks on the subject, or to Cay ley, Messenger of 
Mathematics, 2, 6, 1872. To summarise : 

1. The envelope locus satisfies the original equation but is 
not included in the general solution (see xx', Fig. 112). 




FIG. 112. Nodal and Tac Loci. 

2. The tac locus is the locus passing through the several 
points where two non- consecutive members of a family of curves 
touch. Such a locus is represented by the line AB (Fig. 79), PQ 
(Fig. 112). The tac locus does not satisfy the original equation, 
it appears in the ^-discriminant, but not in the C-discriminant. 




I 126, llo\V TO SOLVE l>l 1'TKKKNTI AL EQUATIONS, 

8, The node locus is the locus passing through the different 
points where each curve of a given family crosses itself (the point 
of intersection node may be double, triple, etc.). The node 
locus does not satisfy the original equation, it appears in the 
C-discriminant but not in the ^-discriminant. BS (Fig. 112) is a 
nodal locus passing through the nodes A, . . ., J5, . . ., C, . . ., .V. 

4. The cusp locus 
passes through all the 
cusps (page 136) formed by 
the members of a family 
of curves. The cusp locus 
does not satisfy the original 
equation, it appears in the & 
p- and in the C-discrimin- FIG. 113. Cusp Locus, 

ants. It is the line Ox in Fig. 113. Sometimes the nodal or 
cusp loci coincides with the envelope locus.* 

EXAMPLES. Find the singular solutions and the nature of the other loci 
in the following equations : 

(1) x 2 !) 2 - 2yp + ax = 0. 

For equal roots y 2 = ax 2 . This satisfies the original equation and is not 
included in the general solution : x 2 - 2Cy + aC 2 0. y 2 = ax 2 is thus the 
singular solution. 

(2) 4xp 2 = (3x - a) 2 . General solution : (x + C) 2 = x(x - a) 2 . 

For equal roots in p, x(3x - a) 2 0, or x(3x - a) 2 = (^-discriminant). 
.For equal roots in C, differentiate the general solution with respect to C. 
Therefore (x + tydxjdC - 0, or C = - a 1 . .-. x(x - a) 2 = (C-discriminant) 
is the condition to be fulfilled when the C-discriminant has equal roots. 
x = is common to the two discriminants and satisfies the original equation 
(singular solution) ; x = a satisfies the C-discriminant but not the ^-dis- 
criminant and, since it is not a solution of the original equation, x = a 
represents the node locus; x = $a satisfies the p- but not the C-discriminaut 
nor the original equation (tac locus). 

(3) p 2 + 2xp - y = 0. 

General solution : (2x :i + 3xy + C) 2 = 4(x 2 + z/) :t ; ^-discriminant : x 2 + y = ; 
C-discriminant : (x 2 + y) s = 0. The original equation is not satisfied by 
either of these equations and, therefore, there is no singular solution. Since 
{x 2 + y) appears in both discriminants, it represents a cusp locus. 

(4) Show that the complete solution of the equation, y' 2 (p* + 1) = a 2 , is 
y 2 + (x - C) 2 = a 2 ; that there are two singular solutions, y = + a ; that 
there -is a tac locus on the x-axis for y = (Fig. 79, see also $ 138). 



* The second part of van der Waals' The Continuity of the Gaseous and 

of Aggregation Binun/ Mi. tin res (1900) has some examples of the preceding 
mathematics". 



304 HIGHEK MATHEMATICS. 12T. 

126. Trajectories. 

This section will serve as an exercise on some preceding work. A trajectory- 
is a curve which cuts another system of curves at a constant angle. If this 
angle is 90 the curve is an orthogonal trajectory. 

EXAMPLES. (1) Let xy = C be a system of rectangular hyperbolas, to 
find the orthogonal trajectory, first eliminate C by differentiation with respect 
to x, thus we obtain, 

xdy/dx + y = 0. 

If two curves are at right angles (^ = 90), then from (17), 32, n- = (a - o) r 
where a, a' are the angles made by tangents to the curves at the point of 
intersection with the a?-axis. But by the same formula, 

tan (+ ?r) (tan a - tan a)/(l + tan a . tan a'). 
Now tan + \ir = oo and 1 / GO = 0, 

.*. tan a - cot o ; or, dy/dx = - dx/dy.* 

The differential equation of the one family is obtained from that of the 
other by substituting dy/dx for - dx\dy. Hence the equation to the orthogonal 
trajectory of the system of rectangular hyperbolas is, xdx + ydy = 0, or 
x 2 - y 2 = C, a system of rectangular hyperbolas whose axes coincide with 
the asymptotes of the given system. 

(2) For polar coordinates show that we must substitute - dr/r . d6 for 
r . de/dr. 

(3) Find the orthogonal trajectories of the system of parabolas y 2 = lax* 
Ansr. Ellipses, 2 2 + y* = C 2 . 

(4) Show that the orthogonal trajectories of the equi potential curves y % 
llr - llr' = C. are the magnetic curves cos + cos 9' = C. 



127. Symbols of Operation. "IJjjt .. W* 

j ',.*'- 

It will be found convenient to denote the symbol of the operation 
" d/dx" by the letter "D". If we assume that the infinitesimal increments 
of the independent variable dx have the same magnitude, whatever be the 
value of aj, we can suppose D to have a constant value. Thus 

d d 2 d A 
AlA-D", , stand for ^^^3" - 

dy d^y d 3 y 
Dy,D*y, . . ., stand for ^, ^, ^, ... 

The operations denoted by the symbols D, D 2 , . . ., satisfy the elementary 
rules of algebra except that they are not commutative f with regard to the 
variables. For example, 

* No doubt the reader sees that in (18), 12, dx/dy is the cotangent of the angle 
whose tangent is dy/dx. 

t The so-called fundamental laws of algebra are : I. The laio of association : The 
number of things in any group is independent of the order. II. The commutative law : 
(a) Addition. The number of things in any number of groups is independent of the 
order, (b) Multiplication. The product of two numbers is independent of the 



I r_'s. im\v TO SOLVE I>MTKI;KM IAI. EQUATIONa ;;:, 

D(u + v +...) = Du + Dv + . . . , (distributive law). 

D(Cu) = CDu, (commutative law), 

where C is a constant. We cannot write D(xy) = D(yx). But, 

h 'iyu - D m + n u (index law), 

is true when m and n are positive integers. If 

Du = v ; u D - l v ; or, u = J~M ; 

.-. v = D.D~ l v, or, D.D- 1 = 1; 

that is to say, by operating with D upon D~ l v, we annul the effect of the 
D- 1 operator. It is necessary to remember later on, that if Dx = 1, 

_1-V -L-. J.^ 

D* - 2 ' D 3 ~ 2 . 3 ' 

In this notation, the equation 

g-<. + * + ,, = 0. 

is written, 

{D 2 - (a + )D + a)8}?/ = ; or, (D - a) (D - fiy = 0. 

Now replace D with the original symbol, and operate on one factor with y. 
Thus, 



By operating on the second factor with the first, we get the original equation 
back again. 



128. The Linear Equation of the nth Order. 

(General Remarks.) 

As a general rule the higher orders of differential equations 
are more difficult of solution than equations of the first order. As 
with the latter, the more expeditious mode of treatment will be to 
refer the given equation to a set of standard cases having certain 
distinguishing characters. By far the most important class is the 
linear equation. 

A linear equation of the nth order is one in which the de- 
pendent variable and its n derivatives are all of the first degree 
and are not multiplied together. The typical form in which it 
appears is 

++-. .+**-*. w 



order. III. The distributive law : (a) Multiplication. The multiplier may be distri- 
buted over each term of the multiplicand, e.g., m(a + b) = ma + mb. (b) Division. 
(a + b)/m = a/m + b/m. IV. The index law ; (a) Multiplication. a'a = + ". 
(b) Division. a m /a n = a m - ". 

U 



306 HIGHER MATHEMATICS. $ 128. 

Or, in symbolic notation, 

Dy + X^*-^ + . . . + X n y = X y 

where X, X v . . . , X n are either constant magnitudes, or func- 
tions of the independent variable x. If the coefficient of the 
highest derivative be other than unity, the other terms of the 
equation can be divided by this coefficient. The equation will 
thus assume the typical form (1). We have studied the linear 
equation of the first order in 123. For the sake of fixing our 
ideas, the equation 

+&*-*> --(*) 

of the second order, will be taken as typical of the class. P, Q, R 
have the meaning above attached to X lt X 2 , X. 

The general solution of the linear equation is made up of two 
parts. 

1. The complementary function which is the most general 
solution of the left-hand side of equation (2) equated to zero, or, 

d*y/dx* + Pdy/dx + Qy = 0. . . (3) 

'The complementary function involves two arbitrary constants. 

2. The particular integral which is any solution of the 
original equation (2), the simpler the better. In particular cases 
when the right-hand side is zero, the particular integral does not 
occur. 

To show that the general solution of (2) contains a general solu- 
tion of (3). Assume that the complete solution of (2) may be 
written, 

y = u + v, . . . . (4) 

where v is any function of x which satisfies (2), that is to say, v is 
the particular integral * of (2), u is the general solution of (3), to 
be determined. Substitute (4) in (2). 

*? + Pp + Qu + p > + P ^ + Q V - B . 

dx 2 dx dx 2 dx 

But . 



x 



. 

dx 



therefore, + P + Qu = 0. 

dx' 2 dx 

Therefore, u must satisfy (3). 

Given a particular solution* of the linear equation, to find the 

* Not to be confused with the particular solution of page 289. 



.< !_".. HOW TO SOLVK DIKKKKKMIAI. K< >r.\TIONS. 

complete solution. Let y = v be a particular solution of the 
following equation, 



where P and Q are functions of x. Substitute y = ur, 
d 2 u f n dv r, \du n 



This equation is of the first order and linear with du/dx as the 
dependent variable. Put dn/dx = z and 

, +(<&. + Pv}z =0; * + 2 dv + Pdx = 0; 

dx \ dx ) z v 

log + 2 log, + \Pdx = 0; or, < = C-'"*. 



2 ; or, y = 

where C l and C 2 are arbitrary constants. 

EXAMPLES. (1) If y = e ax is a particular solution of (Pyldx 1 = dhj, show 



that the complete solution is y = C^"* + C^~ ax . 

(2) If y = x is a particular solution of (1 - x*)d*yldx* - xdy/dx + y = 0, 
the complete solution is y = C l N /(l - x 2 ) + C^x. 

If a particular solution of the linear equation is known, the 
order of the equation can be lowered by unity. This follows directly 
from the preceding result. If y = v is a known solution, then, 
if y = tv be substituted in the first member of the equation, the 
coefficient of t in the result, will be the same as if t were constant 
and therefore zero, t being absent, the result will be a linear equa- 
tion in t but of an order less by unity than that of the given equation. 
It follows directly, that if n particular solutions of the equation are 
known, the order of the equation can be reduced n times. 

For the description of a machine designed for solving (3), see 
Proceedings of the Eoyal Society; 24, 269, 1876 (Lord Kelvin). 

129. The Linear Equation with Constant Coefficients. 

The integration of these equations obviously resolves itself into 
finding the complementary function and the particular integral. 

First, when the second member is zero, in other words, to find 
the complementary function of any linear equation with constant 
coefficients. The typical equation is, 

+ p + g, = o, . . . (i) 



308 HIGHER MATHEMATICS. $ 129. 

where P and Q are constants. The particular integral does not 
appear in the solution. 

If the equation were of the first order, its solution would be, 
y = Ce /tndx . On substituting e mx for y in (1), we obtain 

(m 2 + Pm + Q)e mx = 0, 
provided m 2 + Pm + Q = 0. . . . (2) 

This equation is called the auxiliary equation. If m l be one 
value of m which satisfies (2), then y = e m ^, is an integral of (1). 
But we must go further. 

Case 1. When the auxiliary equation has tiuo unequal roots, 
say m l and m 2 , the general solution of (1) may be written down 
without any further trouble. 

y = C^'-i* + C 2 e m *. ... (3) 

EXAMPLES. (1) Solve (D- + 14Z> - 32) ?/ = 0. Assume ?/ = Ce" 1 * is a 
solution. The auxiliary becomes, m 2 + 14w - 32 = 0. The roots are m 2, 
or - 16. The required solution is, therefore, y = C^ + C 2 e~ x . 

(2) Solve d^y/dx* - m^y = 0. Ansr. y = C^ m * + C. 2 e~ mx (see page 319). 

(3) Show that y = C^ x + C^e x is a complete solution of 

+ 4dyldx + By = 0. 



Case 2. When the two roots of the auxiliary are equal. If 
m l = m 2 , in (3), it is no good putting (C l + C 2 )e m i x as the solution, 
because C l + C 2 is really one constant. The solution would 
then contain one arbitrary constant less than is required for the 
general solution. To find the other particular integral, it is usual 
to put 

m> 2 = m l + h, 

where h is some finite quantity which will ultimately be made 
zero. With this proviso, we write the solution, 
y = Lt h = Q C 1 e^'+ C,e<"'i + *>*. 
Hence, y = Lt h = Q e"^(C l + C 2 e**). 

Now expand e hx by Maclaurin's theorem (page 230). 
/. y = Lt h ^^ x {C l + C 2 (l + hx + ~ 
= Lt h = Q e m ^{C l + 2 + C 2 hx(l + ^, 
= Lt h = ^ x (A + Bx + ^C 2 h 2 x* + C 2 B), 

where R denotes the remaining terms of the expansion of c /ijr , 
A = C l + C 2 , B = C 2 h. Therefore, at the limit, 

y = e"^(A + Bx). . . (4) 

For the sake of uniformity, we shall still write the arbitrary inte- 
gration constants C v C 2 , C 3 , . . . 



g !_".. IfOW TO SOLVE DIFFERENTIAL EQUATION-. :;o<> 

For an equation of a still higher degree, the preceding result 
may be written, 

y = e'^(G l + C& + C.^ + . . . + C r _ t ^f~^. . (5) 
where r denotes the number of equal roots. 

EXAMPLES. (1) Solve d*yjdx* - dPy/dx* - dy/dx + y = 0. Assume 
y = Ce >njc . The auxiliary equation is w 1 - w 2 - m + 1 = 0. The^-roots are 
1, 1, - 1. Hence the general solution can be written down at sight : 
y = C l e~*+(C 9 + C^)e*. 

(2) Solve (D 3 - 3D 2 + 4)y = 0. Ansr. e**(Ci + C^x) + Cy? -'. 

Case 3. When the auxiliary equation has imaginary roots, all 
unequal. Remembering that imaginary roots are always found in 
pairs in equations with real coefficients (page 386), let the two 
imaginary roots be 

m l = a + ifi ; and m^ = a - ifi. 

Instead of substituting y = e nu * in (3), we substitute these values 
of m in (3) and get 



* + C 2 e ~ &) ; 

= e^C^cos fix + i sin fix) + e^C^oos fix - L sin fix). (6) 
(See the chapter on "Hyperbolic Functions ".) Separate the real 
and imaginary parts, as in Ex. 3, p. 280, 

y = e**(C l + C 2 ) cos fix + i(C l - C. 2 ) sin fix ; 
if we put C l + C 2 = A, i(C l - C 2 ) = B, 

y = ex(A cos fix + B sin fix) ..... (7) 
In order that the constants A and B in (7) may be real, the 
constants C l and <7 2 must include the imaginary parts. 

EXAMPLES. (1) Show from (6) that 

y = (cosh ax + sinh our) (A l cos 0x + B l sin &x). 
(Exercise on Chapter VI.) 

(2) Integrate cPyldx* + dy/dx + y = 0. The roots are a = - * and = i >/3 ; 
.-. y = e ~ *l*(A cos \ x/3 . x + B sin \/3 . x). 

(3) The equation of a point vibrating under the influence of a periodic 
force, is, 

//- / t 

^J + a?x = a cos 2xy. 

Find the complementary function. The roots are + 10. From (7) 

y = A cos ax + B sin ax. 
(3) If (D* - IP + D - l)y = 0, y = C l cos x + C a sin x + O*. 

Case 4. When some of the imaginary roots of the auxiliary 
equation are equal. If a pair of the imaginary roots are repeated, 



310 HIGHER MATHEMATICS. $ 130. 

we may proceed as in Case 2, since, when m^ = m 2 , C^e 1 ' 1 ^ + C./'"^, 
is replaced by (A + Bx)e m * x ; similarly, when w 3 = w 4 , C 3 e'"* r + C 4 e'"* r 
may be replaced by (C + Dx)e m * x . If, therefore, 

m l = m. 2 = a + i(3 ; and w 3 = m 4 = a - i/?, 
the solution 

2, = (C T + C 2 x)e( + ^* + (C 3 + C 4 z)e< a - l ^, 
becomes y = e^(^ + Zte) cos /fcc + (C + Dx) sin /to. . (8) 



EXAMPLES. (!) Solve (H - 12Z) 3 + 62D 2 - 156D + 169)?/ = 0. Given the 
roots of the auxiliary : 3 + 2t, 3 + 2t, 3 - 2i, 3 - 2t. Hence, 
T/ = ^{(Cj + C z x) sin 2a; + (C, + C 4 z) cos 2x}. 
(2) If (D 2 + 1) (D - I) 2 ?/ = 0, y = (A + Bx) sin x + (C + Dx} cos x + (E + Fx)e*. 

Second, ivhen the second member is not zero, that is to say to 
find both the complementary function and the particular integral. 
The general equation is, 

3+4+*-* ; 

where P and Q are constant, R is a function of a?. We have just 
shown how to find one part of the complete solution of the linear 
equation with constant coefficients, namely, by putting E, in (9), 
equal to zero. The remaining problem is to find a particular in- 
tegral of this equation. The more useful processes will be described 
in the next section. 

In the symbolic notation, (9) may be written, 

f(D)y = R. . . (10) 

The particular integral is, therefore, 

. (11) 



The right-hand side of either of equations (11), will be found to 
give a satisfactory value for the particular integral in question. 

Since the complementary function contains all the constants 
necessary for the complete solution of the differential equation, it 
follows that no integration constant must be appended to the par- 
ticular integral. 

130. How to find Particular Integrals. 

It will be found quickest to proceed by rule : 

Case 1 (General). When the operator f(D) ~ 1 can be resolved 



130. HOW TO SOL VK DIFFERENTIAL BQUATION& :;ii 

into factors. We have seen that the linear differential equation of 
the first order, 

dyldx - ay = li ; or, y = R/(D - a), . . (1) 
is solved by 

y = e^e ~ '"Rdx ..... (2) 

The term Ce tuc in the solution of (1), belongs to the complementary 
function. 

Suppose that in a linear equation of a higher order, say, 

d*y/dx* - 5dy/dx - Gy = R, 

the operator f(D)~ l can be factorised. The complementary func- 
tion is written down at sight from, 

(> 2 - 5D + G)y = ; or, (D - 3) (D - 2)y = 0, 
namely, y = C^ + C. 2 e 2 ' ..... (3) 

The particular integral is 



i ~ (D - 3) (D - p 

= e**\e-**Rdx - e^e-**Rdx, .... (4) 
from (2). The general solution is the sum of (3) and (4), 

.-. = C + Ce 2 * + e 



EXAMPLES. (1) In the preceding illustration, put R = e** and show that 
the general solution is, C^e 3 * + C.^ 2 * + %e ix . 

(2) If (D* - 4Z> + S)y = 2e**, y = C^* + C#>* + xe 3 *. 

Case 2 (General). When the operator f(D)~ l can be resolved 
into partial fractions with constant numerators. The way to 
proceed in this case is illustrated in the first example below. 

EXAMPLES. (1) Solve d?yldx 2 - 3dy/dx + 2y = e 3 *. In symbolic notation 
this will appear in the form, 

(D - 1) (D - 2)y = e**. 

The complementary function is y = C a e* + C#**. The particular integral is 
obtained by putting 



according to the method of resolution into partial fractions. Operate with 
the first symbolic factor, as above, 

2/j = e**\e - **e**dx - c*je - *<?*dx = $&*. 
The complete solution is, therefore, y CjC* + C^ + \e**. 
(2) Solve (D - 2)*y = x. Ansr. 

Case 3 (Special). When R is a rational function of x, say x". 
This case is comparatively rare. The procedure is to expand 
f(D) ~ l in ascending powers of D as far as the highest power of 
a; in R. 



312 HIGHER MATHEMATICS. 130. 

EXAMPLES. (1) Solve d*y[dx 2 - Idyjdx + y = x z . The complementary 
function is y = e^(A + Bx] ; the particular integral is : 



(2 - D) ~ W = l + 2 + 32 . (2z 2 + 4* + 3). 
(2) If d^y/dx* - y = 2 + 5x, y = C^ + C^e ~ x + 5x - 2. 

Case $ (Special). When B contains an exponential factor, so 
that 

E = e' l *X, 

where X may or may not be a function of x and a has some 
constant value. 

i. When X is a function of x. Since D n e ax = a n e" x , where n is 
any positive integer (page 38), we have (page 25) 

D(e' lx X) = e ax DX + ae' lx X = e nx (D + a)X, 
and generally, as in Leibnitz' theorem (page 49), 
De' lx X = e ax (D + a) n X; 



The operation \D ~ l e tue X is performed (when X is any function of x) 
by transplanting e' lx from the right- to the left-hand side of the 
operator f(D)~ l and replacing D by (D 4- a). This will, perhaps, 
be better understood from the following examples : 

EXAMPLES. (1) Solve d^y/dx 2 - 2dy/dx + y = x z e' lx . The complete solution 
by page 308, is (C l + xCJe* + (D + 2D + 1)- W*. From (5), 



D 2 - 2Z) + 1 (D - 1) (D - 1) 

By rule : & >x may be transferred from the right to the left side of the operator 
provided we replace D by D + 3. 




We get 

as the value of the particular integral. 

(2) Evaluate (D - 1) l e x logx. Ansr. xe x logxle. 

ii. When X is constant. If X is constant, the operation (5) 
reduces to 



The operation f(D)~ l e' lx X is performed by replacing D + a by a. 

EXAMPLES. (1) Find the particular integral in (D 2 - 3D + 2)y - e? x . 

Obviously, 

1 i 

D 2 - 3D + 2* * = 3 2 - 3 . 3 + 2 e * 
(2) Show that %e x , is a particular integral in 

2dy/dx + 1 = e*. 



: < 190. HOW TO SOLVE DI I-TKKKM I A I. K^'ATIONS. 313 

An anomalous case arises when a is a root of f(D) = 0. By this 
method, we should get for the particular integral of dy/dx - y e. 



The difficulty is evaded by using the method (5) instead of (6). 
Thus, 



The complete solution is, therefore, y = Ce* + xe*. 

Another mode of treatment is the following : Since a is a root 
of f(D) = 0, by hypothesis, D - a is a factor of /(>) (page 386). 
Hence, 

f(D) = (D-a)f(D); 

= ^ * f><<* = _ 1 e a * = ^ 

(D-a)-f(D) (D-a)'f(a) 



If the root a occurs r times in f(D) = 0, then D - a enters r 
times into/(D). Therefore, 

1 #. _ 1 1 ,,, _ 1 1 r* _ x r * tx 

J0f -(D-aYF&f -(D-ar'm rf(a)' 

EXAMPLES. Find the particular integrals in, (1) (D + l)~y = e~*. Ansr. 
%x*e~ x . Hint. Replace D by D - 1. e~ x D--\ etc. See page 312. 

(2) (D* - l)y = xe*. Ansr. e r (^ - ^x). Hint. First get e*(D - l)~ l x t 
then e*(l + D + . . .)x, etc. 

Case 5 (Special). When R contains sine or cosine factors. By 
the successive differentiation of sm(nx + a), 

(D' 2 )"sm(nx + a) = ( - ri 2 ) n sm(nx + a).* 
where n and a are constants. 

.-. /(D 2 )sin(na; + a) = f( - n 2 )sm(nx + x). 

(^ + a) = sin(fr + a). (9) 



It can be shown in the same way that, 



(nx + a) = jp^n cos ( 7 ^ + a )' ' ( 10 ) 



EXAMPLES. (1) Find the particular integral of 

d-ty/dx? + d*yjdx* + dyjdx + y = sin 2x. 

Here ' 



* The proof resembles a well-known result in trignometry, 19 : 
D(sin nx) = d(sin nx)/dx = n cos nx ; 
D a (sin nx) = d?(sin nx)/dx* = - n 2 siu nx, etc. 



314 HIGHER MATHEMATICS. 130. 

Substitute for D 2 = (- 2 2 ) as in (9). We thus get - %(D + I)~ 1 sin2a;. Mul- 
tiply by D - 1 and again substitute D 2 = ( - 2 2 ) in the result. Thus 

T i T (/) _ i) s in 2x, or T V(2 cos 2x - sin 2x) 
is the desired result. 

(2) Solve d'ty/dx 2 - k*y = cos mx. Ansr. C^e kx + C 2 <?-**- (cos mx)l(m*+ ft 2 ). 

(3) If o and ft are the roots of the auxiliary equation derived from 

d?y/dt 2 + mdyjdt + n*y = a sin nt, 
(Helmholtz's equation for the vibrations of a tuning-fork) show that 

Cye at + C# st - (acosnt)lmn, 
is the complete solution. 

An anomalous case arises when D' 2 in D 2 + n 2 is equal to - n 2 . 
For instance, the c6mplementary function of d 2 y/dx 2 + n 2 y = cos nx, 
is C^osnx + C 2 smnx, the particular integral is (D' 2 + ri 2 )cosnx. 
If the attempt is made to evaluate this, by substituting D 2 = - ri 2 , 
we get (cos nx)/( - n 2 + ri 2 ) == GO cos nx. We were confronted with 
a similar difficulty on page 243. The treatment is practically the 
same. We take the limit of (D' 2 + n' 2 )cos nx, when n becomes n + h 
and h converges towards zero. 



.'.Zrfc 

= Lt h 
= Lt h 
Lt h 


- n* + n 2 - (n + h)' 2 + n 2 


- (n + /i) 2 + 7i 2 

- n (cos nx . cos lix - 
1 2nh + h^ 

1 f /, 7iV 
- n - J cos nxl 1 H 
1 2nh + W\ \ 2! 

1 fcosnx 


"- 27i/i - W 
sin 710; . sin hx) ; 

+ . . . ) - smnx(hx -...)> 
+ powers of h\ 




27i + h\ h 



But cos nx is contained in the complementary function and hence, when 
h 0, we obtain, 

X + (a term in the complementary function). 

This latter may be disregarded when the particular integral alone is under 
consideration. The complete solution is, therefore, 

y = Cjcos nx + C 2 sin nx + (x sin nx)/2n. 

EXAMPLES. (1) Show that - fyccosx, is the particular integral of 
(D 2 + l)2/ = sin x. 

(2) Evaluate (D 2 + 4) - ^os 2x. Ansr. Jo: sin 2ar. 

(3) Evaluate (D 2 + 4)- 1 sin2. Ansr. %xcos2x. 

(4) Solve dPyJda? - y = x sin x. The particular integral consists of two 
parts, %{(x - 3)cosx - xsinx}. The complementary function is 



But see next case. 

Case 6. TFifeew I? contains some power of x as a factor. Say, 



.1. HOW TO SOLVE DIFFERENTIAL EQUATION :;i:. 

where X is any function of x. The successive differentiation of 
two products gives, $ 20, 

D"xX = xD"X + nD"-*X. 
.-.f(D)xX = xf(D)X + f(D)X. 

Substitute Y = f(D)X, where Y is any function of x. Operate 
with /(.D)- 1 , we get 



EXAMPLES. (1) Find the particular integral in d*yldx* - y = .re 2 *. From 
(11), the particular integral is 



(2) Show in this way, that the particular integral of (/)* - l)y = x sin x, 
is ^(u* 2 cos x - 3x sin a*). 

(3) Solve d?y/dx z - y = av^sin x. 

Ansr. y = C^' + C^ e - ^V*{(10.r + 2)coso.- + (5x - 14)sin x}. 

(4) Integrate d z y/dx 2 - y = 2 cos x. 

Ansr. ?/ = C^e* + C&-* + xsinx + ^cos x(l - x 2 ). 



131. The Linear Equation with Variable Coefficients. 

Case 1. The homogeneous linear differential equation. The 
general type of this equation is : 



where X is a function of ; a lt a 2 , . . . , a n are constants. This 
equation can be transformed into one with constant coefficients by 
the substitution of 

x = e* ; or z = log x. 
we then have, 

dx/dz = e z and therefore, xdy/dx = dyjdz. . (2) 
Just as we have found it very convenient to employ the symbol 

" D," to denote the operation " W' so we shall find it even more 

convenient to denote the operation "#-=-," by the symbol ".9". 

ax 

"9" is treated in exactly the same manner as we have treated 
" D " * in 128 and subsequently. 

* A little care is required in using this new notation. The operations of differentia- 
tion and multiplication by a variable are not commutative. The operation .c 2 /) 2 is not 
the same as $ 2 ,-or as xD . .cD. But we must write, 
xDy =,V//: 



- 2) . . . ( - n 



316 HIGHER MATHEMATICS. g 131. 

EXAMPLES. (1) # = xD = x = d L- 
dx dz 
(2) Show that $x m = mx. 

i. The complementary function. From the first of equations (2), 
we have 12, 9, 

dy_dyd^_ldy d*y _ ^(dfy _ dy\ t 
dx dz ' dx ~ x ' dz ' dx* ~ x\dx* dz) ' 
Substitute these values in (1). The equation reduces to one 
with constant coefficients which may be treated by the methods 
described in the preceding sections. 

EXAMPLES. (1) Solve 

3x . dy/dx - 3y = x* + x. 



'- (3- ~ 

.:y = C^e* + C 2 cos z \/3 + C 3 sin z \ f S + $e 2z + \z&. 
., y = C^x + C 2 cos ( \/3 log x) + C 3 sin ( '\/3 log x) + \x + \x log x. 
(2) Solve x 2 . d 2 yldx* + x . dy/dx + q*y = ; i.e. ($ 2 + q*)y = . 
Ansr. y = C l sin (q log x) + C 2 cos (g log x). 

The linear equation with variable coefficients bears the same 
relation to x, that the equation with constant coefficients does to 
e mx . Hence if x m be substituted for y, the factor x m will divide out 
from the result and an equation in m will remain. The n roots 
of this latter equation will determine the complementary function. 

EXAMPLES. (1) Solve y 3 . d^y/dx* + 2x . dyjdx - 2y -= 0. Put y = x m . 
We get 

m(m - 1) + 2(ra - 1) = ; or, (m + 2) (m - 1) = 0. 

Hence from our preceding results, we can write down the complementary 
function at sight, y C^x + C, z x ~ 2 . 

(2) Solve a- 2 . &y\Ax* + x . dy/dx + 2y = 0. Ansr. y = CJx + 

(3) Find the complementary function in {($ - 1) - 3$ + 
Ansr. y = (G! + C 2 log x)x 2 . 

(4) Integrate {$(3- - 1) - 1\y = 0. Ansr. y = C x x 2 + C^x. 



ii. The particular integral. We may use the operator #, to 
obtain the particular integral of linear equations with variable 
coefficients in the same way that D was used to determine the 
particular integral of equations with constant coefficients. 

The symbolic form of the particular integral is, 

R 



The operator f(3) ~ 1 may be resolved into partial fractions or into 
factors as in the case of D. 



132. HOW TO SOLVE DIFFKKKM I AL K< >r ATION8. :;17 

EXAMPLES. (1) Show that y = C^x 4 + C^jx + ^r 1 log x is a complete 
solution of x a . cPy/dx* - 2x- . dyjdx - 4y = x 4 . 

(2) Find the value of a , Q _ 3^- Using the ordinary method just described 

1 x 3 
we get the indeterminate form g oTT^- In tnis case we must adopt the 

method of page 312 and write 



(3) Solve x z . d*yldx* + 7-r . dyjdx + 5y = x*. Write this 

{3(3 - 1) + 73 + 5}y = x*. 

The particular integral is ( 2 + 6 + 5) - 1 x 5 , or a.- 5 / 60 - The complementary 
function is C^x ~ l + C^x ~ f> . 

(4) Solve x*d?y/dx* + 4x . dyjdx + 2y = e?. 

Ansr. y = CJx + C 2 /u- a + e*jx*. 

(5) Solve ar 5 . d^y/dx 5 + 2x- 2 . d z yjdx 2 - x . dyjdx + y = x + &. 

Ansr. y = CJx + C 2 x + C 3 x log x + }j(log x) 2 + x :i /16. 

(6) Solve x 3 . &yldx* + 2x 2 . d*yldx* + 2y = lOx + W/x. 

Ansr. ?/ = CjX cos (log a-) + C. 2 x sin (log .r) + 5x + C 3 /j; + (2 log x)[x. 

(7) Find the particular integral of the third example in the last set. 
Ansr. x 3 . 

(8) Equate example (2), of the preceding set, to 1/x, instead of to zero, 
and show that the particular integral is then (log x)/x. 

Case 2. Legendre's Equation. Type : 
(a + bx)^ + A l (a+ bx) ~ ^^ + . . . + Ay = R, (3) 

where A lt A. 2 , . . ., A n are constants, R is any function of x. This 
sort of equation is easily transformed into the homogeneous equa- 
tion and, therefore, into the linear equation with constant co- 
efficients. To make the former transformation, substitute z = a + bx, 
for the latter, e = a + bx. 

EXAMPLES. (1) Solve 

(a + bxy . cPyldx 12 + b(a + bx) . dy/dx + c*y = 0. 
Ansr. y = C 1 sin {(c/6) log (a + bx)} + C 2 cos |(c/6) log (a + bx)}. 
(2) Solve (x + a)*d*yldx* - l(x + a) . dy\dx + 6y = x. 
Ansr. y = C^x + a) 2 + C^x + a) 3 + 



132. The Exact Linear Differential Equation. 

A very simple relation exists between the coefficients of an 
exact differential equation which may be used to test whether the 
equation is exact or not. Take the equation, 



318 HIGHER MATHEMATICS. 132. 

where X Q , X v . . ., B are functions of x. Let their successive 
differential coefficients be indicated by dashes, thus X', X", . . . 

Since X Q . d s y/dx z has been obtained by the differentiation of 
X Q 2 . d 2 y/dx 2 , this latter is necessarily the first term of the integral 
of (1). But, 



dx 
Subtract the right-hand side of this equation from (1), 



Again, the first term of this expression is a derivative of 
(X l - X' )dy/dx. This, therefore, is the second term of the in- 
tegral of (1). Hence, by differentiation and subtraction, as before, 

(X, - X\ + JE )g + X 3 y = B. . . (3) 

This equation may be deduced by the differentiation of 
(X 2 - X\ + X 6 ")y, provided the first differential coefficient of 
(X 9 - X\ + X' ] with respect to x, is equal to X s , that is to say, 
" X, - X\ + X'\ = X, ; or, X, - X 2 + X\ - X", = 0. (4) 
But if this is really the origin of (3), the original equation (1) has 
been reduced to a lower order, namely, 



This equation is called the first integral of (1), because the order 
of the original equation has been lowered unity, by a process of 
integration. 

Condition (4) is a test of the exactness of a differential equation. 

If the first integral is an exact equation, we can reduce it, in 
the same way, to another first integral of (1). The process of 
reduction may be repeated until an inexact equation appears, or 
until y itself is obtained. Hence, an exact equation of the nth 
order has n independent first integrals. 

EXAMPLES. (1) Is the equation 

x 5 . d*yldx s + 15z 4 . dhj\dx^ + 60^ . dyjdx + GOx^y = e x exact ? 
From (4), X~ = GOz 2 ; X' z = 180,r 2 ; X'\ = ISOo- 2 ; A"" = 60.r 2 . Therefore, 
X, - X'% + X'\ + X'" Q = and the equation is exact. 

(2) Solve the equation 

x&ylda? + (x 2 - 3)d 2 ijldx* + 4ar . Ay\dx + 2y = 0. 

as far as possible, by successive reduction. The process can be employed 
twice, the residue is a linear equation of the first order, not exact. 

(3) Solve the equation given in example (1). 

Ansr. x*y e x -f C x x 2 + C z x + C 3 . 



gl33. HOW TO SOLVK PII-TKUKNTIAI. Knl A 1 IONS. Wfi 

There is another </ttic& practical test for exact differential r<//ni- 
tinns (Forsyth) which is not so general as the preceding, \\hen 
the terms in X are either in the form of ax m , or of the sum of 
expressions of this type, x'"d"y/dx n is a perfect differential co- 
efficient, if m < n. This coefficient can then be integrated what- 
ever be the value of y. If m = n or m > n, the integration cannot 
be performed by the method for exact equations. To apply the 
test, remove all the terms in which m is less than n, if the re- 
mainder is a perfect differential coefficient, the equation is exact 
and the integration may be performed. 

EXAMPLES. (1) Apply the test to 

X s . d^yjdx 4 + a- 2 . d^y/drf + x . dy/dx + y = 0. 

x . dy[dx + y remains. This has evidently been formed by the operation 
D(xy), hence the equation is a perfect differential. 
(2) Apply the test to 

(x*D* + x*D'* + x 2 D + 2x)y = sin x. 

x* . dyjdx + 2xy remains. This is a perfect differential, formed from D(x' i y). 
The equation is exact. 

If two independent first integrals are known the equation is 
sometimes easily solved. The elimination of dy/dx between two 
first integrals will give the complete solution. 

133. The Integration of Equations with Missing Terms. 

Differential equations with missing letters are common. 
First, the independent variable is absent. Type : 

d*y/dtf = qy ; or, ffiyjda* = qf(y). . (I) 

This equation is, in general, neither linear nor exact. 
Case 1. Whenf(y), in (1), is negative, so that 

5+5% = 0, . . (2) 

where the academic x and y have given place to t and x respectively, 
in order to give the equation the familiar form of the equation of the 
motion of a particle under the influence of a central attracting force. 
Multiply both sides of the equation by %dx/dt, and integrate 
with respect to x, 

dx d 2 x dx 



Separate the variables and integrate again, 

S?/V* Sj* 

ri-f = qdt- cos - 1- = (qt + c), 



320 HIGHER MATHEMATICS. $ 133. 

where e is the integration constant and C = gV 2 . The solution 
involves two arbitrary constants a and e, which respectively denote 
the amplitude and epoch of a simple harmonic motion, whose 
period of oscillation is 2?r/g. Put C l = a cos e, and C 2 = - a sin c. 
x = Cj cos qt + C 2 sin qt. 

Case 2. When f(y) in (1) s positive, the solution assumes the 
form, 

# = C^e 9 * 4- C 2 e ~ qt ; or, x = A cosh qt + B sinh g, 
as on page 309. All these results are important in connection 
with alternating currents and other forms of harmonic motion. 

Another way of treating equations of type (2), occurs with an 
equation like 



V-P, . . . (3) 

which has the form of the standard equation for the small oscilla- 
tions of a pendulum in air. Under this condition, the resistance 
of the air is negligible. Let 

p = dy/dx, .-. d' 2 y/dx 2 = dp/dx = p . dp/dy. 

Substitute these results in the given equation, multiply through 
with 2/?/. 



Multiply by y 2 and 



> , 

where C 4 is an arbitrary constant. The rest is obvious. 

EXAMPLES. (1) The solution of equation (3) is sometimes written in the 
form 

if = C 2 1 sinh(2ic + C 2 ). 
Verify this. 

(2) Solve d 2 x/dt 2 + px + v = 0. Put x = x^ + vj/j. and afterwards omit the 
suffix. Ansr. x = vf/u + Cjcos t \//* + C 2 sin t *fjL 

(3) If the term ^x in the preceding example had been of opposite sign, 
show that the solution would have been, x = vj/j. + CjCosh t \//t + C 2 sinh t \V 
where p. is negative. 

(4) Solve d^y/dx* - a(dy/dx) 2 = 0. Ansr. C^x + C 2 = e. 

(5) Solve 1 + (dy/dx) 2 = yd 2 yjdx z . Ansr. y = cosh(z/a + 6). 

(6) Fourier's equation for tlw propagation of heat in a cylindrical bar, is 
d z Vldx 2 - &V = 0. Hence show that V = C^P* + Ce~^. 

Second, the dependent variable is absent. Type : 

2 = x; or, d*y/da? = f(x). . . (4) 



I i:;:J. HOW TO SOLVE DIFFERENTIAL K^l ATION8. 321 

If these equations are exact, they may be solved by successive 
integration. 

If the equation has the form 

d*y/dx* + dy/dx + x = 0. 

d*v 1 dv P 

Say, j-3 + - 7 = - 7, 

dr 2 r dr Ip' 

a familiar equation in hydrodynamics, it is usually solved by sub- 
stituting p = dy/dx, .-. dp/dx = d 2 y/dx 2 . The resulting equation 
is of the first order, integrable in the usual way. 

EXAMPLES. (1) The above equation represents the motion of a fluid in a 
cylindrical tube of radius r and length I. The motion is supposed to be 
parallel to the axis of the tube and the length of the tube very great in 
comparison with its radius r. P denotes the difference of the pressure at 
the two ends of the tube. If the liquid wets the walls of the tube, the velocity 
is a maximum at the axis of the tube and gradually diminishes to zero at the 
walls. This means that the velocity is a function of the distance (r^ of the 
fluid from the axis of the tube. Solve the equation, remembering that ^ is a 
constant depending on the nature of the fluid. 

Substitute p = dvjdr, 



To evaluate C l in (5), note that at the axis of the tube r = 0. This means 
that if Cj is a finite or an infinite magnitude the velocity will be infinite. 
This is obviously impossible, therefore, C l must be zero. To evaluate C 2 , note 
that when r = r lt v vanishes and, therefore, we get the final solution of the 
given equation in the form, v = ^P(r z l - " 2 )/fyi, which represents the velocity 
of the fluid at a distance r x from the axis. 

(2) Solve ad^y/dx* = \/l + (dyldxf. Make the necessary substitutions and 
integrate. 

a . dp/ \/(l + p 2 ) = dx ; becomes x/a = log(p + >/p* + 1) + C ; 
or, in the exponential form, 



by squaring. On integration 

(3) Some expressions can be reduced to the standard form by an obvious 
transformation. Thus, 

d^yjdx 6 - d^jjdx 3 = x. 
Substitute p for d' A yldx* and differentiate p = dPyJdx 3 twice. Thus, 

dPpjdx 2 - p = x t 
whence y can be obtained by successive integration as indicated above. 

(4) Solve d 2 V/dr* + 2dV/n.dr = 0. This equation occurs in the theory of 

X 



322 HIGHER MATHEMATICS. 134. 

potential. Put dV/dr for the independent variable and divide through. On 
integration 



where log C t is an arbitrary constant. Integrate again 

dV/dr = CJr, becomes V = C 2 - CJr. 
(5) If x . d^y/dx* = 1, show that y = x log x + C-p + C 2 . 

134. Equations of Motion, chiefly Oscillatory Motion. 

By Newton's second law, if a certain mass (m) of matter is 
subject to a constant force (F ) for a certain time, we have, in 
rational units, 

F Q = (Mass) x (Acceleration of the particle). 

If the motion of the particle is subject to friction, we must regard 
the friction as a force tending to oppose the motion generated by 
the impressed force. But friction is proportional to the velocity 
(v) of the motion of the particle, and equal to the product of the 
velocity and a constant called the coefficient of friction, written, 
say, p.. Let F l denote the total force acting on the particle in the 
direction of its motion, 

F l = F - i*.v = md*s/dt*. ... (1) 

If there is no friction, we have, for unit mass, 

F = d 2 s/dt 2 ..... (2) 

The motion of a pendulum in a medium which offers no resist- 
ance to its motion, is that of a material particle under the influence 
of a central force (F) attracting with an intensity which is pro- 
portional to the distance of the particle away from the centre of 
attraction. That is (Fig. 7), 

*--* ..... (3) 

where q- is to be regarded as a positive constant which tends to 
restore the particle to a position of equilibrium the so-called co- 
efficient of restitution. It is written in the form of a power to 
avoid a root sign later on. The negative sign shows that the 
attracting force (F) tends to diminish the distance (s) of the par- 
ticle away from the centre of attraction. If s = 1, q 2 represents 
the magnitude of the attracting force unit distance away. From (2), 

d' 2 s 

&--* ..... (4) 

This is a typical equation of harmonic motion, as will be shown 
directly. One solution of (4) is 

s = C cos(qt + e). . . . (5) 



; l:,t. HOW TO SOLVE DIFFERENTIAL Kol'ATIONS. 323 

This equation is the simple harmonic motion of 50, C denotes 
the amplitude of the vibration. If e = 0, we have the simpler 

equation, 

s = Ccosqt ..... (6) 

When the particle is at its greatest distance from the central 
attracting force, qi = TT, 50, page 112. For a complete to and 
fro motion, 2t = T = period of oscillation, hence 

r.-2x/3. ... (7) 

Equation (4) represents the small oscillations of a pendulum ; 
also the undamped* oscillations of the magnetic needle of a galvano- 
meter. 

In the sine galvanometer, the restitutional force tending to restore the 
needle to a position of equilibrium, is proportional to the sine of the angle of 
deflection of the needle. If J denotes the moment of inertia of the magnetic 
needle and G the directive force exerted by the current on the magnet, the 
equation of motion of the magnet, when there is no retarding force, is 

(8) 



For small angles of displacement, </> and sin <j> are approximately equal. 
Hence, 

3-7* ...... < 9 > 

From (4), q = x/G/J, and therefore, from (9), 

T = 2,rNOyG, ..... (10) 

a well-known relation showing that tlie period of oscillation of a magnet in 
the magnetic field, when there is no damping action exerted on the magnet, is 
proportional to the square root of the moment of inertia of the magnetic needle, 
and inversely proportional to tlie square root of tlie directive force exerted by the 
current on Hie magnet. See page 524. 

In a similar manner, it can be shown that the period of the small oscilla- 
tions of a pendulum suspended freely by a string of length I, is 2ir\ / '//<7, where 
g denotes the acceleration of gravity. 

Equation (4) takes no account of the resistance to which a 
particle is subjected as it moves through such resisting media as 

* When an electric current passes through a galvanometer, the needle is deflected 
and begins to oscillate about a new position of equilibrium. In order to make the 
needle come to rest quickly, so that the observations may be made quickly, some 
resistance is opposed to the free oscillations of the needle either by attaching mica or 
aluminum vanes to the needle so as to increase the resistance of the air, or by bringing 
a mass of copper close to the oscillating needle. The currents induced in the copper 
by the motion of the magnetic needle, react on the moving needle, according to Lenz's 
law, so as to retard its motion. Such a galvanometer is said to be damped. When 
the damping is sufficiently great to prevent the needle oscillating at all, the galvano- 
meter is said to be "dead beat " and the motion of the needle is aperiodic. In ballistic 
galvanometers, there is very much damping. 



324 HIGHER MATHEMATICS. $ 134. 

air, water, etc. This resistance is proportional to the velocity, and 
has a negative value. To allow for this, equation (4) must have 
an additional negative term. We thus get 

d*s ds 

dt* = ~ ^dt ~ qS > 

where /x is the coefficient of friction. For greater convenience, we 
may write this 2/, 



Before proceeding further, it will perhaps make things plainer 
to put the meaning of this differential equation into words. The 
manipulation of the equations so far introduced, involves little more 
than an application of common algebraic principles. Dexterity in 
solving comes by practice. Of even greater importance than quick 
manipulation is the ability to form a clear concept of the physical 
process symbolised by the differential equation. Some of the most 
important laws of Nature appear in the guise of an " unassuming 
differential equation". The reader should spare no pains to acquire 
familiarity with the art. The late Professor Tait has said that 
" a mathematical formula, however brief and elegant, is merely a 
step towards knowledge, and an all but useless one until we can 
thoroughly read its meaning ". 

In equation (11), the term d' 2 s/dt 2 denotes the relative change 
of the velocity of the motion of the particle in unit time, 7 ; 
%f.ds/dt shows that this motion is opposed by a force which 
tends to restore the body to a position of rest, the greater the 
velocity of the motion, the greater the retardation ; q' 2 s represents 
another force tending to bring the moving body to rest, this force 
also increases directly as the distance of the body from the position 
of rest. To investigate this motion further, we cannot do better 
than follow Professor Perry's graphic method. 

The first thing is to solve (11) for s. This is done by the method of 
130. Put s = e mt and solve the auxiliary quadratic equation. We thus 
obtain 

m = - f x/tf 2 - S 2 ) ...... (12) 

And finally, 

s = e - ( + 0X ; or rather, s = C^e - at + C 2 e - ft, 

where a - - / + ^(f - g 2 ) and - - / - v /(/ 2 - 2 2 ). The solution of (11) 
thus depends on the relative magnitudes of / and q. 

Suppose that we know enough about the moving system to be able to 
determine the integration constants. When t = 0, let v = v and s = 0. 



I 184 HOW TO SOLVE IHKFKKKNTIAL EQUATION* 

Case i. Tlie roots of (he (in.>-ill<iri/ equation are real and unequal. The 
condition for real roots - a and - )8, in (12), is that / is greater than q 
(page 388). In this case, 

8 = cy - + c^ - &, . . . . (is) 

solves equation (11). To find what this means, let us suppose that / = 3, 
q = 2, t = 0, s = 0, V Q = 0. From (12), therefore, 

m = - 3 \/9~^~4 = - 3 2-24 = - -76 and - 5-24. 

Substitute these values in (13) and differentiate for the velocity v or ds/dt. 
Thus, 

s = d - 5>24 < + C# - ; ds/dt =- - 5-24CV - 5>24 < + -76C./ - . 

.-. - 5-24CJ + -75C 2 = 1. 
From il3), when t = 0, s - and C l + C 2 = 0, or C v = + C 2 = ,' ; , 

.>.s = l(e-" n <-e-***) (14) 

Assign particular values to t, and plot the corresponding values of s by means 
of Tables XXI. and XXII. Curve No. 1 (Fig. 114) was obtained by plotting 
corresponding values of s and t obtained in this way. 




FIG. 114 (after Perry). 

Case ii. The roots of the auxiliary equation are real and equal. The 
condition for real and equal roots is that / = q. 

.:s = (C l + C 2 t)e-f< (15) 

As before, let / = 2, q = 2, t = 0, s = 0, ?' - 1. The roots of the auxiliary 
are - 2 and - 2. Hence 

s = (C, + C z t)e - * ; and dsfdt = C^e - * - 2(Cj + C 2 t)e - . 

.-. C 2 - 2C a = 1, d = and C 2 = 1 ; or s = te - *. . . (16) 
Plot (16) in the usual manner. Curve 2 (Fig. 114) was so obtained. 

Case iii. The roots of the auxiliary equation are real, equal and of opposite 
ftitjn. For equal roots of opposite sign, say q, we must have/ = 0. Then 

s = Cj sin qt + C 2 cos qt (17) 

Let t = 0, s = 0, v = 1, q = 2, / = 0. Differentiate (17), 
ds/dt = qC l cosqt - qC 2 sing/. 



326 HIGHEK MATHEMATICS. 134. 

Hence 1 = 2C a x 1 - 2 x C 2 x 0, or C a = \ ; .-. C 2 = 0. Hence the equation, 

s = $ sin 2. (18) 

A graph from this equation is shown in curve 4 (Fig. 114). 

Case iv. The roots of the auxiliary equation are imaginary. For imaginary 
roots, - / x /(/2 - q 2 ), or, say a + bt, it is necessary that / < q (page 388). 
In this case, 

s = e- at (Ci sin bt + C 2 cos bt) (19) 

Let the coefficient of friction, / = 1, q = 2, t = 0, s = 0, v - 1. The roots of 
the auxiliary are m = - 1 + \/l - 4 = 1 + \/ - 3 = - 1 + l-7i, where 
t = \/ - 1. Hence a 1, 6 = 1-7. Differentiate (19), 

ds/d = - ae at (C l sin bt + C 2 cos &) + be "'(Cj cos bt - C 2 sin &). 
From (19), C = and, therefore, C l = 1/6 = -57. Therefore, 

s = -Sle-* sml-lt (20) 

Curve 3 (Fig. 114) was plotted from equation (20) in the usual way. 

There are several interesting features about the motions re- 
presented by these four solutions of (11), shown graphically in 
Fig. 114. Curves Nos. 3 and 4 (Cases iv. and iii.) show the 
conditions under which the equation of motion (11) is periodic or 
vibratory. The effects of increased friction due to the viscosity of 
the medium, is shown very markedly by the lessened amplitude 
and increased period of curve 3. The net result is a damped 
vibration, which dies away at a rate depending on the resistance 
of the medium (2/.i;) and on the magnitude of the oscillations 
(q^s). Such is the motion of a magnetic or galvanometer needle 
affected by the viscosity of the air and the. electromagnetic action 
of currents induced in neighbouring masses of metal by virtue of 
its motion ; it also represents the natural oscillations of a 
pendulum swinging in a medium whose resistance varies as the 
velocity. Curve 4 represents an undamped oscillation, curve 3 a 
damped oscillation. 

Curves 1 and 2 (Cases i. and ii.) represent the motion when the 
retarding forces are so great that the vibration cannot take place. 
The needle, when removed from its position of equilibrium, returns 
to its position of rest after the elapse of an infinite time. (What 
does this statement mean? Compare with page 329.) Kaymond 
calls this an aperiodic motion. 

To shoiv that the period of oscillation is augmented by damping. From 
equation (19) we can show that 

s = e ~ at A sin bt (21) 

50. The amplitude of this vibration corresponds to that value of t for 
which s has a maximum or a minimum value. These values are obtained 
in the usual way, by equating the first differential coefficient to zero, hence 

e - at (b cos bt - a sin bt) = (22) 



134. HOW TO SOLVK 1)1 1 FKUKX 1 I.\ L BQUATK 

If we now define the angle <j> such that bt = <j>, or 

tan <t> = bfa ...... 

4>, lying between and (i.e., 90), becomes smaller as a increases in value. 
We have just seen that the imaginary roots of - / x'/ 1 - q 2 are - a 61, 
for values of / less than q. Let 

a* + b- = 2* ...... (-J4) 

The period of oscillation of an undamped oscillation is, by (7), T = 2w/g, 
of a damped oscillation T = 2/6. 

.-. T*IT* =-- 2 2 /6 2 = (rt 2 + 6 2 )/6 2 = 1 + a 2 /6 2 . 

.-. T/T (} = ^(a^-~~b*)lb, .... Uo) 

which expresses the relation between the periods of oscillation of a damped 
and of an undamped oscillation. The period of vibration is thus augmented 
on damping. 

It is easy to show by plotting that tan <p, of (23), is a periodic function 
such that 

tan <p = tan (< + ) = tan (<f> + 2ir) = . . . 
Hence </>, <p + IT, <f> + 2?r, . . . 

satisfy the above equation. It also follows that 

bt v bt 2 + TT, bt ; , + -2ir, . . . 

also satisfy the equation, where t v ^, t s , ... are the successive values of the 
time. Hence 

6*2 = bt^ + TT, 6/ 3 = MJ + 2ir, . . . ; 
... t z = t, + $T, t s = t, + T, . . . 

Substitute these values in (21) and put s ]? s 2> S 3> ... for the corresponding 
displacements, 

.. s l = Ae <u i sin bt^ ; - .s. 2 = Ae "'2 sin bt z ; . . . 

where the negative sign indicates that the displacement is on the negative 
side. Hence 



. . . . (26) 
The amplitude thus diminishes in a constant ratio. Plotting these successive 
values of s and t, we get s 
the curve shown in Fig. 
115. This ratio is called 
the damping ratio, by Kohl- 
rausch ( 4t Dampfungsver- 

haltnis"). _Jt is written ^j |^ \ X^ "V f 

k. The natural logarithm (j 
of the damping ratio, is 
Gauss' logarithmic decre- 
ment, written A (the or- 
dinary logarithm of k, is 
written L). Hence FlG - 115. Damped Oscillation. 

A = log k = aT log e = aT = air/6, . . . (27) 
and from (25), 




Hence, if the damping is small, the period of oscillation is augmented by a 
small quantity of the second order. 



328 



HIGHER MATHEMATICS. 



134- 



The following table contains six observations of the amplitudes of a 
sequence of damped oscillations : 



Observed 









Deflection. 




A. 


L. 


cq 








O7 


1-438 


0-3633 


0-1578 


4.0 








^to 


1-434 


0-3604 


0-1565 


33-5 


1-426 


0-3548 


0-1541 


23-5 


1-425 


0-3542 


0-1538 


16-5 


1-435 


0-3612 


0-1569 


11-5 


1-438 


0-3633 


0-1578 


8 









Meyer, Maxwell, etc., have calculated the viscosity of gases from the rate 
at which the small oscillations of a vibrating pendulum are damped. 

When the motion represented by equation (11) is subject to 
some periodic impressed force which prevents the oscillations 
dying away, the resulting motion is said to be a forced vibra- 
tion. The equation representing such an oscillation is 



Sij + at + a'' -/(*) 

When f(x) = 0, the equation refers to the natural oscillations of a 
vibrating electrical or mechanical system. The impressed force 
is, therefore, mathematically represented by the particular in- 
tegral of equation (29) (see example (3) below). 

The subjoined examples principally refer to systems in harmonic 
motion. 

EXAMPLES. (1) Ohm's law for a constant current is E = RC ; for a 
variable current of C amperes flowing in a circuit with a coefficient of self- 
induction of L henries, with a resistance of R ohms and an electromotive 
force of E volts, Ohm's law is represented by the equation, 

E = RC + L . dC/dt, .... (30) 

where dC/dt evidently denotes the rate of increase of current per second, 
L is the equivalent of an electromotive force tending to retard the current. 

(i.) Wlien E is constant, the solution of (30) has been obtained in a pre- 
ceding set of examples, 

C = EjR + Be ~ /*, 

where B is the constant of integration. To find B, note that when t = 0, 
C = 0. Hence, 

C = E(l - c~ Kt l L )IR ..... (31) 
The second term is the so-called "extra current at make," an evanescent 



* 1:4. HOW TO SOLVE DIFFERENTIAL EQUATIONS. :;_") 

factor due to the starting conditions. The current, therefore, tends to assume 
the steady condition : C = E/R, when t is very great. 
(ii.) When C is an harmonic function of the time, say, 
C = C sin ql ; .-. dC/dt = C n q cos qt. 
Substitute these values in the original equation, 

E = RC Q sin qt + LC q cos qt, 
or. compounding these harmonic motions ( 60), 

E = C x'fl 8 + Z/V . sin (qt + e), 

where tan - *(Lq/R), the so-called lag* of the current behind the 
electro-motive force, the expression *J(R 2 + Lfiq*) is the so-called imped- 
ance. 

(iii.) Wlien E is a function of Hie time, a&y f(t), 

C = Be - M \ L + ^e - ni'-je - Rt l L f(t) . dt, 

where B is the constant of integration to be evaluated as described above. 
(iv.) WJien E is a simple harmonic function of the time, say, 

E = EQ sin qt, 

then, C = Be ~ */* + E sin (qt + e)/ v /(fl 2 + L 2 g 2 ). 

The evanescent term e Rt l L may be omitted when the current has settled 
down into the steady state. (Why ?) 
(v.) WJien E is zero, 

C = Be -MIL. 
Evaluate the integration constant B by putting C = C , when t = 0. 

(2) The relation between the charge (q) and the electromotive force (E) of 
two plates of a condensor of capacity C connected by a wire of resistance R, is 

E = R . dq/dt + q/C, 
provided the self-induction is zero. Solve for q. Show that when 

E = f(t), q = l ) e- tlRC $e- t i R cf(t).dt + Be-"; 
R 

E = 0, q = Q<p- tlRC ; (Q is the charge when t = 0). 

E = constant, q = CE + Be~ tlRC ; 

E = E s'mqt, q = Be~ tlKC + CE(sinqt + RCqcosqt)l(l + R*C*q*). 

(3) The equation of motion of a pendulum subject to a resistance which 
varies with the velocity and which is acted upon by a force which is a simple 
harmonic function of the time, is 



Show that the complementary function is 

x = A cos(qt + e) + Bsin(qt + ). 



* An alternating (periodic) current is not always in phase (or, "in step") with 
the impressed (electromotive) force driving the current along the circuit. If there 
is > -It'-induction in the circuit, the current lags behind the electromotive force ; if 
there is a condensor in the circuit, the current in the condensor is greatest when the 
electromotive force is changing most rapidly from a positive to a negative value, that 
is to say, the maximum current is in advance of the electromotive force, there is then 
said to be a lead in the phase of the current. 



330 HIGHER MATHEMATICS. .$135. 

To solve the equation, assume that 

x = A cos(qt + e) + B sin(g + e), 
is a solution. Substitute in the given equation, 

.-. - A<- + 2fBq + n*A = !;.-.- Aq* + 2fBq + n*B = 0. 
4 = n * ~ g 2 B = _ 2/g 

( H 2 _ g '2) + 4^2 ' ( W 2 _ g 2) + 4 y2 g 2- 

Put 4 = E cos 6. B = R sin e. 

The solution of the given equation is then 

x = Rcoa(qt + e - ej ), .... (32) 
where R = I/ ^{(n 2 - g 2 )' 2 + 4/ 2 g' 2 } ; tan 6 = 2/#/(>t 2 - g' 2 ). 

The forced oscillations due to the impressed periodic force, are thus de- 
termined by (32). The complementary function gives the natural vibrations 
superposed upon these. 

(4) If the friction in the preceding example, is zero, 

** + n*x = cosfe* + e ) (33) 

A particular integral is x = {f.cos(qt + e)}j(n 2 - q 2 ). This fails when n = q. In 
this case, assume that x = Ctsin(nt + e) is a particular integral. (33) is 
satisfied provided C = fj2n. The physical meaning of this is that when the 
pendulum is acted on by a periodic force " in step " with the oscillations of 
the pendulum, the amplitude of the forced oscillations will increase pro- 
portionally with the time, until, when the amplitude exceeds a certain limit, 
equation (33) no longer represents the motion of the pendulum. 

(5) When an electric current, passing through an electrolytic cell, has 
assumed the steady state, show that the ionic velocity is proportional to the 
impressed force (electromotive force). By Newton's law, for a moving body, 

(Impressed force) = (Mass) x (Acceleration). 

Friction is to be regarded as a retarding force acting in an opposite direction 
to the impressed force ; this frictional force is proportional to the velocity of 
the body. 

.-. (Impressed force less friction) = (Mass) x (Acceleration). 
Express these facts in symbolic language. See (1) above. Integrate the 
result and evaluate the constant for v = 0, when t 0. 

For ionic motion, m is very small, p. is very great. When t is great, show 
that the exponential term vanishes, and 

Ohm's law. Compare with (31). 

135. The Velocity of Simultaneous and Dependent 
Chemical Reactions. 

While investigating the rate of decomposition of phosphine 
( 88), we had occasion to point out that the action really takes 
place in two stages : 

STAGE I. PH 3 = P + 3H. 

STAGE II. 4P = P 4 ; 2T = H 2 . 



g 185, HOW TO soLVK DI I'FKIM-M I A I. EQUATION 

The former change alone determines the velocity of the whole 
reaction. The physical meaning of this is that the speed of the 
reaction which occurs during stage II., is immeasurably faster 
than the speed of the first. Experiment quite fails to reveal the 
complex nature of the complete reaction.* 

Suppose, for example, a substance A forms an intermediate 
compound B, and this, in turn, forms a final product C. If the 
speed of the reaction 

A = B, is one gram per , o<jW second, 
when the speed of the reaction 

B = 6 T , is one gram per hour, 
the observed " order " of the complete reaction 

A = (7, 

will be fixed by that of the slower reaction, B = 6', because the 
methods used for measuring the rates of chemical reactions are not 
sensitive to changes so rapid as the assumed rate of transformation 
of A into B. Whatever the " order " of this latter reaction, B = C 
is alone accessible to measurement. If, therefore, A = C is of the 
first, second, or nth order, we must understand that one of the 
subsidiary reactions (A = B, or B = C) is 

(1) an immeasurably fast reaction, accompanied by 

(2) a slower measurable change of the first, second or nth 
order, according to the particular system under investigation. 

If, however, the velocities of the two reactions are of the same 
order of magnitude, the "order" of the complete reaction will not 
fall under any simple type ( 88, 89), and, therefore, some changes 
will have to be made in the differential equations representing the 
course of the reaction. Let us study some of the simpler cases. 

Case i. In a given system, a substance A forms an intermediate 
substance B, which finally forms a third substance C. 

Let one gram molecule of the substance A be taken. At the end of a cer- 
tain time t, the system contains x of A, y of B, z of C. The rate of diminution 
of x is evidently 

--M ...... '" 



* Professor Walker illustrates this by the following analogy (" Velocity of Graded 
Reactions," Proc. Royal Soc. Edin., Dec., 1897): "The time occupied in the trans- 
mission of a telegraphic message depends both on the rate of transmission along the 
conducting wire and on the rate of the messenger who delivers the telegram ; but it is 
obviously this last, slower rate that is of really practical importance in determining the 
total time of transmission ". . . . 



332 HIGHER MATHEMATICS. 135. 

where k denotes the velocity constant of the transformation of A to B. The 
rate of formation of C is 



where k. 2 is the velocity constant of the transformation of B to C. Again, 
the rate at which B accumulates in the system is evidently the difference in 
the rate of diminution of x and the rate of increase of z, or 

^k.x-kM ...... (3) 

The speed of the chemical reactions, 

A = B = C, 

is fully determined by this set of differential equations. When the relations 
between a set of variables involves a set of equations of this nature, the result 
is said to be a system of simultaneous differential equations. 

In a great number of physical problems, the interrelations of the variables 
are represented in the form of a system of such equations. The simplest class 
occurs when each of the dependent variables is a function of the independent 
variable. 

The simultaneous equations are said to be solved when each variable is 
expressed in terms of the independent variable, or else when a number of 
equations between the different variables can be obtained free from differential 
coefficients. 

To solve the present set of differential equations, first differentiate (2), 



Add and subtract k^y, substitute for dy/dt from (3) and for k 2 y from (2), we 
thus obtain 



But from the conditions of the experiment, 

x + y + z = 1, .-. z - I = - (x + y). 
Hence, the last equation may be written, 



0. '. . (4) 



This linear equation of the second order with constant coefficients, is to be 
solved for z - 1 in the usual manner ( 130). At sight, therefore, 

a - \ = Ctf-** + C#-** ..... (5) 

But 2 = 0, when t = 0, 

.'. C, + C 2 = - 1 ...... (6) 

Differentiate (5). From (2), dz/dt = 0, when t 0. Therefore, making the 
necessary substitutions, 

- Cfa - C 2 k. 2 = ...... (7) 

From (6) and (7), 

C, = kjfa - fc,) ; C 2 = kjfa - kj. 1 
The final result may therefore be written, 

fcg-e-aat + fe * g-*i*. ... (8) 



135. HOW TO SOLVE DIFFERENTIAL K< >I A IIONS. :;:;:; 



Harcourt and Esson have studied the rate of reduction of potassium per- 
manganate by oxalic acid. 

2KMnO 4 + 3MnSO 4 + 2H 2 O = K^SO 4 + 2H t SO 4 + 5MnO 9 ; 
MnO., + HtSO 4 + # 2 C 2 O 4 = MnSO 4 + 2# 2 O + 2CO 2 . 

By a suitable arrangement of the experimental conditions this reaction 
may be used to test equations (5) or (8). 

Let x, y, z, respectively denote the amounts of Mn z O 7 , MnO^ and Mm) 
(in combination) in the system. The above workers found that C, = 28'5 ; 
C 2 = 2-7 ; c-*i = -82 ; e-*2 = -98. The following table places the above sup- 
positions beyond doubt. 





2-1. 




z- 1. 


t 




1 




Minutes. 


Found. 


Calculated. 


Minutes. 


Found. 


Calculated. 


0-5 


25-85 


25-9 


3-0 


10-45 


10-4 


1-0 


21-55 


21-4 


3-5 


8-95 


9-0 


1-5 


17-9 


17-8 


4-0 


7-7 


7'8 


2-0 


14-9 


14-9 


4-5 


6-65 


6-6 


2-5 


12-55 


12-5 


5-0 


5-7 


5-8 



Case ii. A solution contains a gram molecules of each of A and 
C, the substance A gradually changes to B, which, in turn, reacts, 
with C to form another compound D. 

Let x denote the amount of A which remains untransformed after the 
elapse of an interval of time t, y the amount of B, and z the amount of C 
present in the system after the elapse of the same interval of time t. Hence 
show that 

-^ = M; ~m = k ^' (9) 

The rate of diminution of B is proportional to the product of its active mass y 
into the amount of C present in the solution at the time /, but the velocity of 
increase of y is equal to the velocity of diminution of x, 

y = k x k iiz (\Q\ 

If x, y, z, could be measured independently, it would be sufficient to solve 
these equations as in case i., but if x and y are determined together, we must 
proceed a little differently. Note z = x + y. From the first of equations (9), 
and (10) by addition and the substitution of dt = - dx/k^x from (9), and of 

z - x = y, we get 



dz K 



(11) 



where K has been written in place of k^k r The solution of this equation 
has been previously determined (page 298) in the form 

- log a- + Kx - 



334 



HIGHER MATHEMATICS. 



135. 



In some of Harcourt and Esson's experiments, C 1 = 4-68 ; fe 1 = -69 ; k z = -006364. 
From the first of equations (9), it is easy to show that x = ae-h*. Where 
does a come from ? What does it mean ? Hence verify the third column in 
the following table : 





z. 


t 




Minutes. 


Found. 


Calculated. 


2 


51-9 


51-6 


3 


42-4 


42-9 


4 


35-4 


35-4 


5 


29-8 


29-7 



After the lapse of six minutes, the value of x was found to be negligibly 
very small. The terms succeeding log x in (12) may, therefore, be omitted 
without committing any sensible error. Substitute x ae W in the re- 
mainder, 

^(d - log a + \t)z = 1 ; or (C\ + t}z = 1 



where C\ = C^k^ - (loga)/^. Harcourt and Esson found that C\ = O'l, and 
l//c 2 = 157. Hence, in continuation of the preceding table, these investigators 
obtained the results shown in the following table. The agreement between 
the theoretical and experimental numbers is remarkable. 





z. 




z. 


t 




t 




Minutes. 


Found. 


Calculated. 


Minutes. 


Found. 


Calculated. 


6 


25-7 


25-7 


10 


15-5 


15-5 


7 


22-2 


22-1 


15 


10-4 


10-4 


8 


19-4 


19-4 


20 


7-8 


7'8 


9 


17-3 


17-3 


30 


5-5 


5-2 



The theoretical numbers are based on the assumption that the chemical 
change consists in the gradual formation of a substance which at the same 
time slowly disappears by reason of its reaction with a proportional quantity 
of another substance. 

This really means that the so-called " initial disturbances " in chemical 
reactions, are due to the fact that the speed during one stage of the reaction, 
is faster than during the other. The magnitude of the initial disturbances 
depends on the relative magnitudes of fc x and fc 2 . The observed velocity in 
the steady state depends on the difference between the steady diminution 
- dx/dt and the steady rise dzjdt. If /c 2 is infinitely great in comparison with 
fcj, (8) reduces to 

z = a(\ - e-W), 



135. HOW TO SOLVE DIFFERENTIAL KnlATIONS. 335 

which will be immediately recognised as another way of writing the familiar 
equation 



So far as practical work is concerned, it is necessary that the solutions of 
the differential equations shall not be so complex as to preclude the possibility 
of experimental verification. 

Case iii. In a given system A combines with R to form B, B 
combines with E to form C, and C combines with E to form D. 

In the hydrolysis of triacetin, 

CjHg . A 3 + H. OH = 3 A . H + C 3 H 6 (OH) 3 , 
(Triacetin) (Glycerol) 

where A has been written for CH 3 . COO . , there is every reason to believe 
that the reaction takes place in three stages : 

C S H 6 .A 3 + H.OH=A.H + C 3 H 6 .A^.OH (diacetin) ; 

C 3 H 5 . Z 2 . OH + H . OH = A . H + C 3 H 5 . ~A . (0#) 2 (monacetin) ; 
C 3 # 5 . A . (OH)* + H . OH = A . H + C 3 H K . (OH) 3 (glycerol). 
These reactions are interdependent. The rate of formation of glycerol is con- 
ditioned by the rate of formation of monacetin ; the rate of monacetin depends, 
in turn, upon the rate of formation of diacetin. There are, therefore, three 
simultaneous reactions of the second order taking place in the system. 

Let a denote the initial concentration (gram molecules per unit volume) 
of triacetin, b the concentration of the water ; let x, y, z, denote the number 
of molecules of mono-, di- and triacetin hydrolysed at the end of t minutes. 
The system then contains a - z molecules of triacetin, z - y, of diacetin, 
y - x, of monacetin, and b - (x + y + z) molecules of water. The rate of 
hydrolysis is therefore completely determined by the equations : 

dx/dt = k^y - x) (b - x - y - z) ; . . . (13) 
dyldt = k^(z - y} (b - x - y - z) ; . . . (14) 
dzfdt = k 3 (a - z) (b - x - y - z) ; . . . (15) 

where fc lt fc 2 , fc 3 , represent the velocity coefficients ($ 88) of the respective 
reactions. 

Geitel tested the assumption : k t = 7c 2 = k 3 . Hence dividing (15) by (13) 
and by (14), he obtained 

dz/dy =(a- z)l(z - y} ; dzfdx = (a - z)l(y - x). . . (16) 
From the first of these equations, 

dy + yJL. = ^, 

a - z a - z 

which can be integrated as a linear equation of the first degree. The constant 
is equated by noting that if a = 1, z = 0, y = 0. The reader might do this as 
an exercise on 123. The answer is 

y = z + (a - z)\og(a - z) ..... (17) 

Now substitute (17) in the second of equations (16), rearrange terms and inte- 
grate as a further exercise on linear equations of the first order. The final 
result is, 

x = z + (a - z)\og(a - z) - ?LZ_f{log(a - z)}*. . . (18) 



336 HIGHER MATHEMATICS. 136. 

Geitel then assigned arbitrary numerical values to z (say from O'l to TO), 
calculated the corresponding amounts of x and y from (17) and (18) and com- 
pared the results with his experimental numbers. For experimental and 
other details the original memoir must be consulted (vide infra). 

EXAMPLE. Calculate equations analogous to (17) and (18) on the supposi- 
tion that fcj 4= k 2 4= k s . 

A study of the differential equations representing the mutual conversion 
of red into yellow, and yellow into red phosphorus, will be found in a paper 
by Lemoine in the Annales de Chimie et de Physique [4], 27, 289, 1872. 

There is also a series of interesting papers by Rud. Wegscheider bearing 
on this subject in Zeit. f. phys. Glum., 30, 593, 1899 ; ib., 34, 290, 1900 ; ib., 35, 
513, 1900; Monatsliefte filr Chemie, 22, 749, 1901. 

The preceding discussion is based upon papers by Harcourt and Esson, 
Phil. Trans., 156, 193, 1866; Geitel, Journ. filr prakt. Chem. [2], 55, 429, 
1897; J. Walker, Proc. Roy. Soc. Edin., 22, 1897. It is somewhat surprising 
that Harcourt and Esson's investigation has not received more attention from 
the point of view of simultaneous and dependent reactions. The indispens- 
able differential equations, simple as they are, might perhaps account for this. 
But chemists, in reality, have more to do with this type of reaction than any 
other. The day is surely past when the study of a particular reaction is 
abandoned simply because it " won't go " according to the stereotyped velocity 
equations of 88. 



136. Simultaneous Differential Equations. 

By way of practice it will be convenient to study a few more 
examples of simultaneous equations. 

For a complete determination of each variable there must be 
the same number of equations as there are independent variables. 
Quite an analogous thing occurs with the simultaneous equations 
of ordinary algebra. 

I. Simultaneous equations with constant coefficients. The 
methods used for the solution of these equations are analogous 
to those employed for similar equations in algebra. The opera- 
tions here involved are chiefly processes of elimination and sub- 
stitution, supplemented by differentiation or integration at various 
stages of the computation. The use of the symbol D often shortens 
the work. Most of the following examples are from results proved 
in the regular textbooks on physics. 

EXAMPLES. (1) Solve dxjdt + ay = 0, dyjdt + bx = 0. Differentiate the 
first, multiply the second by a. Subtract and y disappears. Hence writing 
ab ra 2 , 

x = C^ nt + C#~ mt ; or, y = C 2 v/6/a . e ~ mt - Cj *Jbfa . e" lt . 



; I:K,. HOW TO SOLVE DIFFKKKM I A I. EQUATIONS. 337 

We might have obtained an equation in //, and substituted it in the second. 
Thus four constants appear in the result. But one pair of these constants 
can be expressed in terms of the other two. Two of the constants, there- 
fore, are not arbitrary and independent, while the integration constant is 
arbitrary and independent. It is always best to avoid an unnecessary 
multiplication of constants by deducing the other variables from the first 
without integration. The number of arbitrary constants is always equal 
to the sum of the highest orders of the set of differential equations under 
consideration. 

(2) Solve dxjdt + y = 3* ; dyldt - y = x. Differentiate the first. Sub- 
tract each of the given equations from the result. (D 2 - 4D + 4)z remains. 
Solve as usual, x = (C l + C^e*. Substitute this value of x in the second 
of the given equations and y = (C l - C z + C^e*. 

(3) The equations of rotation of a particle in a rigid plane, are 

dxjdt = py ; dyjdt = /JLX. 

To solve these, differentiate the first, multiply the second by /u, etc. Finally 
x = C l cos ij.t + C z sin p.t ; y = C\ cos /*t + C' 2 sin fit. To find the relation 
between these constants, substitute these values in the first equation and 

- fj.C l sin ^t + /j.C<z cos pt = pC\ cos fit + /j.C'% sin pi, 
or C\ = - C' 2 and C 2 = C\. 

(4) Solve tfxIdP = - n*x d*yldt z = - n*y. 

x = C l cos nt + C 2 sin nt ; y = C' 2 cos nt + C' z sin nt. 
Eliminate / so that 

(C\x - C$Y + (C'ax - C 2 7/) 2 - (CjC' 2 - C^) 2 , etc. 

The result represents the motion of a particle in an elliptic path, subject to a 
central gravitational force. 

(5) Solve dxjdt + by + cz = ; dyjdt + a^x + c^z + ; dzldt + a z x + b z y=0. 
Operate on the first with Z> 2 - 6 2 c l5 on the second with 6 2 c - bD, on the third 
with bc^ - cD. Add. The terms in y and z disappear. The remaining 
equation has the integral, 

x = C^ + C# Bt + C#*\ 
where o, , 7, are the roots of 

& - (a^b + 2 c + b^cjz + a^b^c + a^bc^ = 0. 

The values of y and z are easily obtained from that of x by proper substi- 
tutions in the other equations. 

(6) If two adjacent circuits have currents ^ and i,, then, according to the 
theory of electromagnetic induction, 



(see J. J. Thomson's Elements of Electricity and Magnetism, p. 382), where 
-Rj, R 2 , denote the resistances of the two circuits, L ly L%, the coefficients of 
self-induction, E lt E z , the electromotive forces of the respective circuits and 
M the coefficient of mutual induction. All the coefficients are supposed 
constant. 

First, solve these equations on the assumption that E l = E z = 0. Assume 
that 

ii = ae and ij = be*, 
Y 



338 HIGHER MATHEMATICS. 136 

satisfy the given equations. Differentiate each of these variables with respect 
to t and substitute in the original equation 

aMm + b(L z m + R 2 ) = ; bMm + ft(L a w + #,) = 0. 
Multiply these equations so that 

(L^ - M 2 )w 2 + (L^z + R^m + R^ 2 = 0. 

For physical reasons, the induction L^L Z must always be greater than M. 
The roots of this quadratic must, therefore, be negative and real (page 388), 
and 

i x = a^e ~ '"i f , or a 2 e - '"2* ; i 2 = \e - '"i, or b# - '"2*. 
Hence, from the preceding equation, 

a-^Mm-^ + b l L 2 m l + B^ - ; or a^ = (L^m^ + R z )IMm l ; 
similarly, 2 /^2 Min 2 [(L l m 2 + -^i)- 

Combining the particular solutions for ^ and i 2 , we get 

ij = a-^e - m i f + a 2 e ~ '"a* ; i 2 = \e - m i* + b 2 e - m tf, 
the required solutions. 

Second, if E-^ and E 2 have some constant value, 



are the required solutions. 

II. Simultaneous equations with variable coefficients. The 
general type of simultaneous equations of the first order, is 
P,dx + QJy + R,dz = ; 
P 2 dx + Q. 2 dy + R 2 dz = 0, . . 
where the coefficients are functions of x, y, z. These equations 
vcan often be expressed in the form 

dx _ dy _ dz 
~P ~ Q ~ ~R' ' 

which is to be looked upon as a typical set of simultaneous equa- 
tions of the first order. If one of these equations involves only 
two differentials, the equation is to be solved in the usual way, 
and the result used to deduce values for the other variables, as in 
the first of the subjoined examples. 

When the members of a set of equations are symmetrical, the 
solution can often be simplified by taking advantage of a well- 
known theorem * in algebra (ratio). According to this, 

* Perhaps it is best to state the proof. Let 

da-IP = dy/Q = dz\R = &, say ; then, 

dx = Pk ; dy = Qk ; dz = Rk ; 
or, Idx = IPh mdy = mQk ; ndz = nRk. 
Add these results, 

Idx + mdy + ndz = k(lP + mQ + nR). 

Idx + mdy + ndz , _ dx dy _ dz 

' TP + mQ + nR~ = = ~P = ~Q = ~R' 



<i:;7. HOW TO SOLVE l>l l-'KKKKNTIAI. EQUATIONS. :;:;-. 

dx dy dz Idx + mdy + ndz _ I'dx + m'dy + n'dz 
~P = ~Q = ~R = IP + mQ + nR '' " IP + m'Q + n'H == " ( 3 ) 
where I, m, n, I', m', ri, . . . are sets of multipliers such that 

IP + mQ + nR = ; IT + m'Q + n'R = ; . . . (4) 

hence, Idx + mdy + ndz = 0, etc. . . . (5) 

The same relations between x, y, z, that satisfy (5), satisfy (2). 

If (4) be an exact differential equation, equal to say du, direct 

integration gives the integral of the given system, viz., 

u = a, (6) 

where a denotes the constant of integration. 
In the same way, if 

Idx + mdy + ndz = 0, 

is an exact differential equation, equal to say dr, then, since dv is 
also equal to zero, 

r-6. . . . . (7) 

is a second solution. These two solutions must be independent. 

EXAMPLES. (1) Solve dx[y = dy/x = dz\z. The relation between dx and 
dy contains x and y only, the integral, t/ 2 - x 2 = C lt follows at once. Use 
this result to eliminate x from the relation between dy and dz. The result is 

dz\z = dyJJ(y* - CJ ; or, y + x % 2 + C\) = C^z. 

These two equations, involving two constants of integration, constitute a 
complete solution. 

(2) Solve dx/(mx - ny) = dyf(nx - Iz) dz/(ly - mx). I, m, n and or, y, z 
form a set of multipliers satisfying the above condition. Hence, 

Idx + mdy + ndz = ; xdx + ydy + zdz 0. 
The integrals of these equations are 

u = Ix + my + nz C 1 ; r = x' 2 + y 2 + z 2 = C 2 , 
which constitute a complete solution. 

(3) Solve dx/(x 2 - y 2 - z*) = dyfixy = dzftxz. From the two last equa- 
tions y = Cz. Substituting x, y, z for I, m, n, each of the given ratios is 
equal to 

(xdx + ydy + zdz)\(& + y* + z 2 ). .-. x 2 + 7/ 2 + z* = C^z, 
is another solution. 

137. Partial Differential Equations. 

Equations obtained by the differentiation of functions of three 
or more variables are of two kinds : 

1. Those in which there is only one independent variable, 
such as 

Pdx + Qdy + Rdz = Sdt, 

which involves four variables three dependent and one inde- 
pendent. These are called total differential equations. 



\ 

340 HIGHER MATHEMATICS. 137. 

2. Those in which there is only one dependent and two or 
more independent variables, such as, 

where z is the dependent variable, x, y, t the independent variables. 
These equations are classed under the name partial differential 
equations. 

The former class of equations are rare, the latter very common. 
We shall confine our attention to partial differential equations. 

In the study of ordinary differential equations, we have always 
assumed that the given equation has been obtained by the elimina- 
tion of constants from the original equation. In solving, we have 
sought to find this primitive equation.* Partial differential equa- 
tions, however, may be obtained by the elimination of arbitrary 
functions of the variables as well as of constants. 

It can be shown from Euler's theorem (page 56) that if 

x n f(y z \ 

\x x y 

be a homogeneous function, 



where the arbitrary function has disappeared. f Again, if 

u = f(ax 3 + by 3 ), 
is an arbitrary function of x and y. 

7)U ^)U ^U ^U 

= af(ax 3 + by 3 ) ; = bf(ax 3 + by 3 ) ; .-. b - a = 0. 



* Physically, the differential equation represents the relation between the de- 
pendent and the independent variables corresponding tp an infinitely small change in 
each of the independent variables. 

The reader will, perhaps, have noticed that the term " independent variable " is 
an equivocal phrase. (1) If u =f(z), u is & quantity whose magnitude changes when 
the value of z changes. The two magnitudes u and z are mutually dependent. For 
convenience, we fix our attention on the effect which a variation in the value of z has 
upon the magnitude of u. If need be we can reverse this and write z =f(u), so that 
u now becomes the ''independent variable". (2) If v = f(x, y), x and y are "inde- 
pendent variables " in that x and y are mutually independent of each other. Any 
variation in the magnitude of the one has no effect on the magnitude of the other, x 
and y are also "independent variables " with respect to v in the same sense that z has 
just been supposed the "independent variable" with respect to u. 

f This is usually proved in the textbooks in the following manner : 
Let u = x n f(yjx, z[x, . . . ). Put yfx = Y, z/x = Z, . . . 

.-. 3r/3z = - 0/o-a, -dZfdx = - z/x* . . . ; ^Y/^y = l/x, -dZfdy = 0, . . . 
Let v =f(Y, Z, . . . ), for the sake of brevity, therefore, since u x", 



$ 138. HOW TO SOLVE DIFFERENTIAL EQUATIONS. 341 

EXAMPLES. (1) If y - bu = f(x - an), aJ^ + 6^- = 1. 

(2) If I/* - Ifx = f(lfy - I/a:), y*d*fcx + y*dz[dy = #. 

(8) If z = a(r + y) + 6, 'dzfdx - 'dzfty = 0. 

For this reason an arbitrary function of the variables is added 
to the result of the integration of a partial differential equation 
instead of the constant hitherto employed for ordinary differential 
equations. 

If the number of arbitrary constants to be eliminated is equal 
to the number of independent variables, the resulting differential 
equation is of the first order. The higher orders occur when the 
number of constants to be eliminated, exceeds that of the inde- 
pendent variables. 

If u = f(x, y), there will be two differential coefficients of the 
first order, namely, 'bu/'tix and ~%ufiy ; three of the second order, 
namely, 'b-ufix*, Wufixby, Wufiy' 2 . . . 



138. What is the Solution of a Partial Differential 
Equation ? 

Ordinary differential equations have two classes of solutions 
the complete integral and the singular solution. Particular 
solutions are only varieties of the complete integral. Three 
classes of solutions can be obtained from some partial differential 
equations, still regarding the particular solution as a special case 
of the complete integral. These are indicated in the following 
example. 

The equation of a sphere in three dimensions is, 

aj + y* + *s = r-, . . (1) 

when the centre of the sphere coincides with the origin of the 



by the method for the differentiation of a function of a function, 6 and 9, 
Therefore, 



^ ^ ^ Y & ^ '&>'dz .'do. 

3t/ = ^BT'^ =x " dY'Vz= xn dZ ; te = ' 32' 

Now multiply by x, y, z, . . . respectively, and add, 



342 HIGHER MATHEMATICS. 138 

coordinate planes and r denotes the radius of the sphere. If the 
centre of the sphere lies somewhere on the xy-pl&ne at a point 
(a, b), the above equation becomes 

( X - a) 2 + (y - b)^ + z* = r 2 . . . (2) 

When a and b are arbitrary constants, each or both of which may 
have any assigned magnitude, equation (2) may represent two 
infinite systems of spheres of radius r. The centre of any mem- 
ber of either of these two infinite systems (called a double infinite 
system) must lie somewhere on the xy-pl&ne. 
Differentiate (2) with respect to x and y. 

x _ a + z ^ z ; = o ; y - b + * = 0. (3) 

to ty 

Substitute for x - a and y - b in (2). We obtain 

^ 2 |/^\ 2 /^y +1 | = r2 
\\toy v*y/ / 

Equation (2), therefore, is the complete integral of (4). By 
assigning any particular numerical value to a or b, a particular 
solution of (4) will be obtained, such is 

(x - I) 2 + (y - 79) 2 + z* = r 2 . . . (5) 
If (2) be differentiated with respect to a and b, 

(/ \2 / Z*\2 I "2 9) . (/ \2 / 7A2 2 2) 

^a ' ^>b 

or, # - a = 0, and y - b = 0. 

Eliminate a and 6 from (2), 

* = r, (6) 

This result satisfies equation (4), but, unlike the particular solution, 
is not included in the complete integral (2). Such a solution of 
the differential equation is said to be a singular solution. 

Geometrically, the singular solution represents two plane sur- 
faces touched by all the spheres represented by equation (2). The 
singular solution is thus the envelope of all the spheres represented 
by the complete integral. If AB (Fig. 79) represents a cross sec- 
tion of the #?/-plane containing spheres of radius a, CD and EF 
are cross sections of the plane surfaces represented by the singular 
solution. 

If the one constant is some function of the other, say, 

a = b, 
(2) may be written 

(x - ay + (y - a) 2 + z* = r 2 . . . (7) 



138. How TO SOLVE DIFFEBENTIAL !: i A I IONS. 

Differentiate with respect to a. We find 

a = (x + y). 
Eliminate a from (7). The resulting equation 



is called a general integral of the equation. 

Geometrically, the general integral is the equation to the 
tubular envelope of a family of spheres of radius r and whose 
centres are along the line x = y. This line corresponds with the 
axis of the tube envelope. The general integral satisfies (4) and 
is also contained in the complete integral. 

Instead of taking a b as the particular form of the function 
connecting a and b, we could have taken any other relation, say 
a = \b. The envelope of the general integral would then be like 
a tube surrounding all the spheres of radius r whose centres were 
along the line x = ^y. Had we put a 1 - b- = 1, the envelope 
would have been a tube whose axis was an hyperbola x' 2 - y- = 1. 

A particular solution is one particular surface selected from the 
double infinite series represented by the complete solution. A 
general integral is the envelope of one particular family of surfaces 
selected from those comprised in the complete integral. A singular 
solution is the complete envelope of every surface included in the 
complete integral.* 

Theoretically an equation is not supposed to be solved com- 
pletely until the complete integral, the general integral and the 
singular solution have been indicated. In the ideal case, the 
complete integral is first determined ; the singular solution ob- 
tained by the elimination of arbitrary constants as indicated above ; 
the general integral then determined by eliminating a and f(a). 

Practically, the complete integral is not always the direct ob- 
ject of attack. It is usually sufficient to deduce a number of 
particular solutions to satisfy the conditions of the problem and 
afterwards to so combine these solutions that the result will not 
only satisfy the given conditions but also the differential equation. 



* The study of Gibbs' "Surfaces of Dissipated Energy," " ,s'///yi/vx <</' /> 
tion," " Su/ faces of Cliemical tiqui/i/irit'iu," as well as van der Waals' Surfaces," is 
the natural sequence of 68, 126 and the present section. But to enlarge upon this 
subject would now cause a greater digression than is here convenient. Airy's little 
book, An I-lli:iiii'ntui-ii Treatise on Partial Differential Equation*, will repay careful 
study in connection with the geometrical interpretation of the solutions of partial 
differential equations. 



344 HIGHER MATHEMATICS. 139. 

Of course, the complete integral of a differential equation 
applies to any physical process represented by the differential 
equation. This solution, however, may be so general as to be of 
little practical use. To represent any particular process, certain 
limitations called limiting conditions have to be introduced. 
These exclude certain forms of the general solution as impossible. 
See examples at the end of Chapter VIII. ; also example (1) last 
set 133, and elsewhere. 

The more important varieties of partial differential equations 
from the point of view of this work are the linear equations of the 
second and higher orders. 



139. The Solution of Partial Differential Equations of the 

First Order. 

For the ingenious general methods of Lagrange, Charpit, etc., 
the reader will have to consult the special textbooks, say, For- 
syth's A Treatise on Differential Equations (Macmillan & Co., 
1888). 

There are some special types classified by Forsyth in the 
following order : 

Type I. The variables do not appear directly. The general 
form is, 

fQzfix, tz/ty) = 0. . . . (I.) 
The solution is 

z = ax + by 4- C, 
provided a and b satisfy the relation 

f(a, b) = 0, or b = f(a). 
The complete integral is, therefore, 

z = ax + yf(a) + C. (I) 

EXAMPLES. (1) Solve (9*/3a;) 2 + (ds/dy)* = m 2 . The solution is 

z ax + by + C', 

provided a 2 + 6 2 = m 2 . The solution is, therefore, z = ax + y v /(m 2 - a 2 ) + C. 
For the general integral, put C = f(a). Eliminate a between the two equations, 

' z = ax + ^/(w 2 - a?)y + f(a) ; and x - a/ N /(w 2 - a?)y + f(a) = 0, 
in the usual way. 

(2) Solve pq 1. Ansr. z ~ ax + y/a + /(a). 

NOTE. We shall sometimes write, for the sake of brevity, 
'dzj'dx = p ; 'ds/'dy = 3. 

(3) Solve a(p + q) = z. Sometimes, as here, when the variables do ap- 
pear in the equation, the function of x, which occurs in the equation, may 



i< i :,'.. H<>\v T<> soi.VE DIFFKIlKM I.\l. BQUATIONft :ur, 

be associated with 'dzfdx, or a function of y with 'dzfdy, by a change in the 
variables. We may write the given equation ap\z + aqfz = 1. Put dz\n = d'A ; 
dy/a = dr, dx/a = dX, hence, 9#/3 Y + 'dZftX = 1, the form required. 

(4) Solve x*p* + 2/V = *- Pu t X = log x, y = logy, Z = log z. Proceed 
AS before. Ansr. 2 = Cx a y V(i- 2 ). 

If it is not possible to remove the dependent variable z in this 
way, the equation will possibly belong to the following class : 

Type II. The independent variables x and y are absent. The 
general form is, 



Assume as a trial solution, that 
?>zfiy = a . 

Let t)z/D# be some function of z obtained from II., say p = <j>(z). 
Substitute these values in 

dz = p . dx + q . dy. 

We thus get an ordinary differential equation which can be readily 
integrated. 

dz = <f>(z) . dx + a<f>(z) . dy. 

.'. x + ay = \dz/<f>(z) + C. . . . (2) 

EXAMPLES. (1) Solve p z z + g 2 = 4 Here, 

(a 2 + z) (dzldxY = 4. >J(a? + z) . dzjdx = 2, 

.-. x + C = /^(a 2 + z) . dz = $(a 2 + z) 3 / 2 . Ansr. 2(a 2 + zf = 3(x + ay + C) 2 . 
(2) Solve p(l + q 2 ) = q(z - a). Ansr. 4C(s - a) = (x + Cy + 6) 2 . 

If z does not appear directly in the equation, we may be able 
to refer the equation to the next type. 

Type III. z does not appear directly in the equation, but x and 
dzfix can be separated from y and ^z/^y. The leading type is 

/ 1 (4te/te)-/&,>/*f). . - (HI.) 

Assume as a trial solution, that each member is equal to an 
arbitrary constant a, so that ^zj^x and *bz/ty can be obtained in 
the form, 

Ttzftx = ^(x, a) ; *bzfiy = <f> 2 (y, a). 

dz = p . dx + q . dy, 
then assumes the form 

dz^f l (x,a)dx+/ 2 (y,a)dy. . . (3) 

EXAMPLES. Solve the following equations : 

(1) q - p = x - y. Put 'dz/'dx - x = 'dzf'dy - y = a. Write 

'dzl'dx = x + a, etc. ; 'dz/'dy = y + a. 
Hence, z = $(x + a) 2 + $(y + a) 2 + C. 

(2) g 2 + ^ 2 = x + y. Ansr. z = \(x + a)*P + ;{(,/ - a)** + C. 

(3) q = 2?#> 2 . Ansr. z = ax + a-// 2 + <-' 



346 HIGHER MATHEMATICS. 140. 

Type IY. Analogous to Clairaut's equation. The general 
type is 

z = p . x + q . y + f(p . q). . (IV.) 

The complete integral is 

z = ax + by + f(a, b). . . . (4) 

EXAMPLES. Solve the following equations : 

(1) z = px + qy + pq. Ansr. z = ax + by + ab. Singular solution 2= -xy. 

(2) z = px + qy + k J(l + p* + g 2 ). Ansr. z = ax + by + k \/l + a 2 + 6 2 . 
Singular solution, x 2 + y* + z 2 = r 2 . The singular solution is, therefore, a 
sphere ; r, of course, is a constant. 

(3) z = px + qy - n *Jpq. Ansr. z = ax + by - n sjab. Singular solution, 



140. Partial Differential Equations of the nth Order. 

These are the most important equations that occur in physical 
mathematics. There are no general methods for their solution, 
and it is only possible to perform the integration in special cases. 
The greatest advances in this direction have been made with the 
linear equation. Before proceeding to this important equation, it 
appears convenient to solve some simpler types. 

EXAMPLES. Integrate the following equations : 

(1) o -p, = a. If^pr =P', ^ = Integrate with regard to y and we 

get p = ay + f'(x). It is very possible that/'() is a function of y. Integrate 
with respect to x and z = {{ay + f'(x)}dx = axy + f^x) + f 2 (y). 

9% x 

( 2 ) 3^- ~ y = a. Ansr. z = Jx 2 log y + axy + f^x) + f 2 (y). 

HcM = %) Ansr.* = S[e- 

There are many points of analogy between the partial and the 
ordinary linear differential equations. Indeed, it may almost be 
said that every ordinary differential equation between two variables 
is analogous to a partial differential in the same class. The solu- 
tion is in each case similar, but with these differences : 

First, the arbitrary constant of integration in the solution of 
an ordinary differential equation is replaced by a function of a 
variable or variables. 

Second, the exponential form, Ce" 1 *, of the solution of the 



ordinary linear differential equation assumes the form e 






in. HOW TO sni.YK l>l I-TKKKXTIAL K< >l.\ I InN- 



The expression, e 6y <t>(y), is known as the symbolic form of Taylor's 
theorem. Having had considerable practice in the use of the symbol of 

operation D for ^-, we may now use D' to represent the operation =s- 
By Taylor's theorem, 



where .< is regarded as constant. 

The term in brackets is clearly an exponential series (page 230) equivalent to 

mjc- 3 

e y , or, writing D' f r ^T 

<t>(y + >n.v) = e' llxD 'ty(ij}. .... (1) 
The general form of the linear equation is, 

where A^, A lf . . ., A, may be constants, or functions of x and y. 
As with ordinary linear equations, 

Complete Solution = Particular Integral + Complementary Function. 

The complementary function is obtained by solving the left- 
hand side of equation (2), equated to zero. We may write (2) in 
symbolic form, 

(AJP + A^DU + A. 2 D f ' 2 + A 3 D + Ajy + A.^z = 0, . (3) 

where D is written for ^- ; D' ion: ^- ; DD' for s~^~- Sometimes 



we understand 

^D,D>-0. . . (4) 

in place of (2). 

141. Linear Partial Equations with Constant Coefficients. 
A. Homogeneous equations. Type : 

. d' 2 Z . ~ti' 2 Z . cl~2 



where R is a function of x. 

To find the complementary function, put R = 0, and instead 
of assuming, as a trial solution, that y = e mc , as was the case with 
the ordinary equation, suppose that 

z = <l>(y + 1) ( 6 ) 



348 HIGHER MATHEMATICS. 141. 

is a trial solution. Differentiate (6), with respect to x and y, we 
thus obtain, 

* ^Z Wz 

*) ; = f'(y + mx) ; = mf"(y + m*) ; 



w = m f (y 

Substitute these values in equation (5) equated to zero, and divide 
out the factor f"(y + mx). The auxiliary equation, 

A m 2 + A^m + A 2 = 0. . . . (7) 

remains. If w is a root of this equation, f"(y + mx) = 0, is a 
part of the complementary function. If a and ft are the roots of 
(7), then 

z = ^^(y) + *^ 2 (y), ... (8) 
as in (3), 130. From (6), therefore, 

= A(?/ + oo?) + / 2 (y + #u) . . . (9) 
since a and ft are the roots of the auxiliary equation (7), we can 
write (5) in the form, 

(D + aD') (D + ftD')z = 0. . . . (10) 

EXAMPLES. Solve the following equations : 

(1) %& ~ ^ 2 = - Ansr - z = MV + ^ + MV - x )- 

(2) fjj - 4^ + 4^ 2 - 0. Ansr. ^ . My - 2aj) + / 2 (y - 2*). 

~ 2 = - Ansr - * = 



(4) ^2"= 2 ^2- Ansr. M = Mat + x) + f z (at - x). This most important 

equation, sometimes called d'Alembert's equation, represents the motion of 
vibrating strings, the law for small oscillations of air in narrow tubes (organ 
pipes), etc. 

We cannot say much about the undetermined functions f-^at + x) and 
f z (at - x) in the absence of data pertaining to some specific problem. Con- 
sider a vibrating harp string, where no force is applied after the string has 
once been put in motion. Let x I denote the length of the string under a 
tension equal to the weight of a length L of the same kind of string. In 
order to avoid a root sign later on, a 2 has been written in place of gL, where 
g represents the constant of gravitation. Further, let u represent the dis- 
placement of any part of the string we please, and let the ordinate of one end 
of the string be zero. Then, whatever value we assign to the time t, the 
limiting conditions are u 0, when x = ; and u = 0, when x = L 

.-. f^at) + M^) = ; M^ + I) + f z (at - I) = 0, 

are solutions of d'Alembert's equation. From the former, it follows that 
fi(at) must always be equal to - / 2 () ; 
Q -Mat- Z)=0. 



141. HOW TO SOLVE DIFFERENTIAL EQUATIONS. :u 

r.ut <if may have any value we please. In order to fix our ideas, suppose that 
nt - I -= y t .: tit + I = q + 21, where q has any value whatever. 



The physical meaning of this solution is that when q is increased or diminished 
by 21, the value of the function remains unaltered. Hence, when at is in- 
creased by 21, or, what is the same thing, when t is increased by 21 fa, the 
corresponding portions of the string will have the same displacement. In 
other words, the string performs at least one complete vibration in the time 
2//o. Hence, we conclude that d'Alembert's equation represents a finite 
periodic motion, with a period of oscillation 2l/a. 

EXAMPLE. Show that 

HMat + l) + Mat - 1)} = 0, 
is a solution of d'Alembert's equation, and interpret the result. 

A further study of d'Alembert's equation would require the introduction 
of Fourier's series, Chapter VIII. 

When two of the roots are equal, say a = ft. We know that the 
solution of 

(D - afz = 0, is z = e^C^x + C 2 ), 130 ; 
by analogy, the solution of 

(D - aDJz = 0, is z = 6">/iM + My)}, 
or, z = xj\(y + ax) + f z (y + ax). . . (11) 

EXAMPLES. Solve : 
(1) - 



(2) (D 3 - 3IW + D-D' 2 + D 3 )* = 0. 

Ansr. z = xf^y - x} + f z (y - x) + f 3 (y + x). 

The particular integral will be discussed after. 
B. Non-homogeneous equations. Type : 



. A A 

^ + ^ + ^ + ^ = 0. (12) 

If the non-homogeneous equation can be separated into factors, 
the integral is the sum of the integrals corresponding to each sym- 
bolic factor, so that each factor of the form D - mD', appears in 
the solution as a function of y + mx, and every factor of the form 
D - mD' - a, appears in the solution in the form z = e^ffy + mx). 



Factors, (D + D') (D - D' + l)z = 0. Ansr. z = My - x) + e ~ *My + x). 
Factors, (D + 1) (D - D')z = 0. Ansr. e = e~ ^(y) + MX - y). 



350 HIGHER MATHEMATICS. 141. 

It is, however, not often possible to represent the solutions of 
these equations in this manner. When this is so, it is customary 
to take the trial solution, 

z = e** + fiy t . . . . (13) 

and substitute for z in the given equation (12). Then, 

T>z T>z Wz 

= az . = ftz ; _ - = aftz ; 



Equate the resulting auxiliary equation to zero. We thus obtain 
(A^ + A ia ft + A. 2 (3' 2 + A. 3 a + A 4 ft + A 6 )z = 0. (14) 

This may be looked upon as containing a bracketed quadratic in 
a and (3. For any value of ft, we can find the corresponding value of 
a, or the value of a, for any assigned value of ft. There is thus an 
infinite number of particular solutions of this differential equation. 

If u 1} u 2 , u 3 , . . . , are particular solutions of any partial dif- 
ferential equation, each solution can be multiplied by an arbitrary 
constant and the resulting products are also solutions of the equation. 

Similarly 2 it is not difficult to see that the sum of any number 
of particular solutions will also be a solution of the given equation. 

It is usually not very difficult to find particular solutions, even 
when the general solution cannot be obtained. The chief difficulty 
lies in the combining of the particular solutions in such a way, 
that the conditions of the problem under investigation are satisfied. 
Plenty of illustrations will be found at the end of the next chapter. 

If the above quadratic is solved for a in terms of ft, and if the 
resulting /(a, ft), is homogeneous, we shall have the roots in the 
form, 

a = Wj/3, a = m 2 ft, . . . , a = mJ3. 

The equation will, therefore, be satisfied by any expression of the 
form, 

z = 2C* + > f .... (15) 

where m has any value m 1} m 2 , . . . and C may have any value 
C lf C 2 , . . . The symbol " S " indicates the sum of the infinite 
series, obtained by giving m and C all possible values. 

The above solution (15), may be put in a simpler form when ft 
is a linear function of a, say, ft = aa + b. This applies to equation 
(12). Again, we can sometimes solve the equation z = e ax + ^y = 0, 
for a, in terms of ft. In order to fix these ideas, let us proceed to 
the following examples. 



I u-_'. HOW TO SOLVK I>I FFKKKM I A I. EQUAXION& :;.-,! 

EXAMPLKS. (1) Solve (D 2 -D>=0. Herea 3 -/3=0. Hence, * = CV 
Put a = , a = 1, a = 2, . . . and we get the particular solutions 



Now the difference between any two terms of the form e ax + ^y, is included in 
the above solution, it follows, therefore, that the first differential coefficient 
of e ax + fty t is also an integral, and, in the same way, the second, third and 
higher derivatives must be integrals. Thus we have the following particular 
solutions : 



y = ^ x +2a y) + Qy( x + 2at/)}, etc. 
If a = 0, we get the special case, 

z = <V + C 2 (.r 2 + 2y) + C 3 (a-> + 6xy) + . . . 



(a - /3) (a + jS - 3) = 0. .'. /3 = a and = 3 - a. 
Hence z = *{* + > + ^2C a e a <- r ~ - 7/ ) ; z = f,(y + x) + e^f 2 (y - x). 

(3) S 1Ve ^y + a ^c + b !j, + abz = ' Ansr ' z = e ~ Vf ^ + e ' bXf ^ (y} - 

(4) Solve (D 2 - Z)' 2 + D + 3D' - 2)z = 0. Ansr. z = ej^y -x) + e~ ^f 2 (y + x). 



142. The Particular Integral of Linear Partial Equations. 

The following methods for finding the particular integral of 
homogeneous or non-homogeneous equations, are deduced by 
processes analogous to those employed for the particular integrals 
of the ordinary equations. 

The complementary function of the ordinary linear equation 

(D - m)z = 0, is, Ce mx ; 
so, for the partial equation 
(D - mD')z = 0, we have, e'"* D '<t>(y), or the equivalent <f>(y + mx). 

This analogy extends to the particular integrals. In the former 
case, the particular integral of 

(D - m)z = R, is, z = (D - m)~ l E ; 
while for F(D, D')z = E, we have, z = F(D, D')~ 1 B. 

Case 1 (General). When F(D, D') can be resolved into factors, 
so that, 

z = (D - mD')f(x, y). . (16) 

It is now necessary to find a value for this symbol (16). 
First show that 

De nx E = (D + a)R, 



352 HIGHER MATHEMATICS. 142, 

by putting mD in place of a, and f(x, y), in place of E (Case 4 r 
131)- 



-T/fr *-). . . (17) 

The value sought. The particular integral may, therefore, be 
found by the following series of operations : 

(1) Subtract mx from y in the f(x, y), to be operated upon. 
(2) Integrate the result with respect to dx. (3) Add mx to y, 
after the integration. 

If there is a succession of factors, the rule is to be applied to 
each one seriatim, beginning on the right. 

EXAMPLES. Find particular integrals in the following examples. It is 
well to be careful about the signs of the different terms added and subtracted. 
It is particularly easy to err by want of attention to this. 



Now xy becomes x(y - ax). This, on integration with respect to dx, becomes 
%x*y - %ax z , and finally typy + \av? - \ax*. Hence, 

1 1 _ 1 x*y ax s 

D - aD' ' D + aD' ' D - aD' " ~2~ + IT' 

Subtract - ax from y, for %x"*(y + ax) + ^ax 3 . Integrate and add - ax to the 
result. \a?y remains. This is the required result. 
(2) (D 2 + 3DD' + 2Z>') ~ l x + y. Ansr. \x*y - \x*. 

Case 2 (Special). When E has the form f(ax + by). Multiply 
F(D, D> by D n and get 

D n <f>(D'/D)z = f(ax + by). 
Operate on ax + by with D' and D respectively, 

jjf(ax + by) = -. 

As on page 313, the particular integral is 
" 1 1 



How to use this formula will appear from the examples. 

EXAMPLES. Find particular integrals in, (1) (D* + DD' - 2D' 2 )z 
The particular integral is 

sin(x + 2u) = - ffsinte + 
1 + 2 - 8 JJ 



(2) (D 2 + 5DD' + QD' z )z = lj(y - 2x). Ansr. xlog(y - 



< 14'J. HOW TO SOLVE DIFFF.U.M I A I. KM!' \ 1 h>\-. ;;;,;; 

The above process cannot be employed when F(D, D'), or 
F(a, b) has the same form as R, because a vanishing factor then 
appears in the result. In such a case, use the above method for 
all factors which do not vanish when a is put for D, b for D. The 
solution is then completed by means of the formula : 



EXAMPLE. Evaluate the particular integral in 

(D - D') (D + 2D')z = x + y. 
For the first factor, use the above method and then 



Case 3 (Special). When E has the form of sin(ax + by), or 
cos(ax + by). Proceed as on page 313, when 

+ fy) = _ -^hXoz + fy), (19) 



F(D*> DD', D' 2 ) F( - a*, - ab, - 6 2 ) 

and in the same way for the cosine. 

EXAMPLES. Find the particular integrals in : 
(1) (D 2 + DD' + IT - l)z = sin (x + 2y). 

sin * + = ' sin 



D* + DD' + D' - I = - 1 - 2 + D' - 

D' + 4 1 

= D' 2 - 16 = " 10> cos ( x + %) + 2 s i n ( x + 2 2/)l- 

(2) (D + DD' - 2D')z = sin (x - y) 4- sin (x + ij). Find the particular 
integral for sin (x - ?/), then for sin (x + y). Add the two results together. 
Ansr. sin (x - y) + $x cos (a + y). 

For the anomalous case proceed as in 131. 



Case 4 (Special). When E has the form e ax + b *, proceed as 
directed on page 312, 



that is to say, put a for D and b for D'. 

EXAMPLES. Find particular integrals in the following : 

(1) (D* - DD' - 2D' 2 + 2D + 2D> = e 2 * + 3 . 

Ansr. = (D 2 - DD' - 2Z)' 2 + 2D + 2Z>') - l e** + 3 * ; = 

(2) (DD' + aD + bD' + ab)z = e m + *. Ansr. e m + **l(m + a) (n + b). 

If F(a, b) = 0, proceed as on page 313, 



354 HIGHER MATHEMATICS. 143. 

where F' a or F' b denotes the first differential coefficient with respect 
to the subscript. The two results agree with each other. 

EXAMPLE. Solve (Z> 2 - D' 2 - 3D + 3D'}z = e*+**. 

Ansr. fax + y) + e s *f 2 (y - x) - ye* + 2 *. 

Case 5 (Special). When B has the form x r y*, where r and s are 
positive integers. Operate with -^(.D, D') ~ l on x r y s expanded in 
ascending powers of r and s. 

EXAMPLES. (1) Find the particular integral in : 
(D 2 + DD' + D - \)z = x*y. 



= - x*y -2y-2 X -x*-. 
The expansion is not usually carried higher than the highest power of the 
highest power in f(x, y). 

(2) Evaluate (D 2 - D' 2 - 3D + 3D') ~ l xy. Ansr. 



(3) (D 2 - a*D'*)z = x. 

1 / D' 2 



Case 6 (Special). When R has the form e ax + by X, where X is a 
function of x or y. Use 

F(D, D') ~ l e ax + ^X = e ax + by F(D + a, D + b)~ 1 X, (22) 
derived as on page 315. 



EXAMPLE. Find the particular integrals in 

xe ax + 



x = 



- D' 



143. The Linear Partial Equation with Variable 
Coefficients. 

These may sometimes be solved by transforming them into a 
form with constants. E.g., 

~ti r + s z 

(i.) Any term x r y s ^ x ^ . may be reduced to the form with con- 
stant coefficients, by substituting u = log x, v = log y. 
EXAMPLES. Solve the equations : 

(i) - y -y + * = a This reduces to ^ 2 ^ u2 ~ ^/ 3t;2=0 - 



1 144. HOW TO SOLVE DIFFERENTIAL EQUATIONS. :,:,:, 

e tin- solution of this equation, z = ^(u + v) + $ 2 (u - v), must be re- 
ted into the form in x and y, thus, z = f^xy) 



+ 2xy &y + 7/ e = ' Ansr ' * = fl(ylx) + xf * (ylx} ' 

+ y) J|/- a |r = a Put 3* = ".a*- 

Ansr. z = /,(y) + J(a? + y) a f' 2 (x) . do-. 
(ii.) The transformation may be effected by substituting 

d 
,9 = a^- and $' = y^, and treating the result as for constant co- 

efficients. 

EXAMPLES. (1) Solve the first two examples of the preceding set in this 
way. 



nz = 0. 



144. The Integration of Differential Equations in Series. 

When a function can be developed in a series of converging 
terms, arranged in powers of the independent variable, an ap- 
proximate value for the dependent variable can easily be obtained. 
The degree of approximation attained obviously depends on the 
number of terms of the series included in the calculation. The 
older mathematicians considered this an underhand way of getting 
at the solution but, for practical work, it is invaluable. As a 
matter of fact, solutions of the more advanced problems in physical 
mathematics are nearly always represented in the form of an ab- 
breviated infinite series. Finite solutions are the exception rather 
than the rule. 

EXAMPLES. (1) Evaluate the integral in f(x) = 0. Assume that/(jr) can 
be developed in a converging series of ascending powers of a% that is to say, 

f(x) = a + a^x + OyK 2 + ayX 3 + ...... (1) 

By integration 

jf(x)dx = f(o + a^x + atfc'* + . . ,)dx ; 

=ja Q dx + ja^xdx + Jo^dz + , . . ; 



. . .) + C. . . . cJi 

(2) It is required to find the solution of dyfdx = y, in series. Assume 
that // = /(a-), has the form (1) above, and substitute in the given equation. 

(i - o) + (2aa -a l )x + (3a 3 -ajx+ . . . =0. . . (3) 



350 HIGHER MATHEMATICS. 144. 

This equation would be satisfied, if 

a - a ' a - 1 a --^a a - l - 1 

2 2 3 2 3 ! 

Hence, y = a Q <f>(x), 

where <j>(x} = 1 + x + x* + x 3 + . . . = e x . 

2 ! 3 ! 

Put a for the arbitrary function, 

.-. y = ae x . 

That this is a complete solution, is proved by substitution in the original 
equation. 

Write the original equation in the form 

where v is to be determined. Hence, 

since <j>(rc) satisfies the original equation, 

dy/dx = 0, or v is constant. 

For equations of higher degree, we must proceed a little differently. For 
example : 

(3) Solve g - x^ - cy = x> (4) 

(i.) The complementary function. As a trial solution, put y = ax. The 
auxiliary equation is 

m(m - I)a a' w - 2 - (m + c)x m = 0. . . . (5) 
This shows that the difference between the successive exponents of x in 
the assumed series, is - 2. The required series is, therefore, 

y = a Q x m + a-fl? - 2 + . . . + a n _ ^x m + * - 2 + a n x m + *, 
which is more conveniently written 

In order to completely determine this series, we must know three things 
about it. Namely, the first term, the coefficients of x and the different powers 
of x that make up the series. 

Substitute (6) in (4), 

h2-a _ ( m + 2n + c)a n x m + 2n = 0, (7) 



~o x 

where n has all values from zero to infinity. If x is a solution of (4), the 
coefficient of x m + 2n ~ 2 must vanish with respect to m. Hence by equating 
the coefficient of x m + Zn ~* to zero,* 

(m + Zn) (m + Zn - l)a n - (m + 2n - 2 + c)a n _ l = 0. . (8) 
If n = 0, m = 0, or m = 1. 

When n is greater than zero, 

m + Zn - 2 + c m \ 



(m + 2n) (m + 2n - 1)" 
This formula allows us to calculate the relation between the successive 
coefficients of x by giving n all integral values 1, 2, 3, ... 

* If we take the other part of the auxiliary a diverging series is obtained, useless 
for our purpose. 



< n:,. Ho\V TO soi.VK mi'I'T.UKNTlAI, EQUATION-. :;:,: 
['irst, suppose m = 0, then we can easily calculate from |'J). 



c + 2 c(c + 2) 

'"* = -37r a ' = "- : 

Next put m = 1, and, to prevent confusion, write 6, in (9), in place of . 
b _ c + 2u - 1. 

" 2n(2n + 1) *~ 1 ' 
proceed exactly as before to find successively 6 lt 6 2 , & . . . 

"hi ''"' ^ 

The complete solution of the equation, is the sum of series (10) and (11), 
or if Ufa = y', fa/ 2 = y", 

y = ay\ + fa/a. 

which contains the two arbitrary constants a and 6. 

(ii.) The particular integral. By the above procedure we obtain the com- 
plementary function. For the particular integral, we must follow a somewhat 
similar method. E.g., equate (8) to # 2 instead of to zero. The coefficient of 
m - 2, in (5), becomes 

m(m - l)a Q x"* ~ 2 = a; 2 ; 
.-. m - 2 = 2 and m(m - I)a = 1 ; 



Substitute successive values of n = 1, 2, 3, . . . in the assumed expansion, 
and we obtain 

(Particular integral) = a^x + o 1 x' r ' + 2 + a,^c m + 4 + . . . , 
where , a v 0%, . . . and m have been determined. 
(4) Solve dtyldx 2 + xy = 0. 

Ansr. y = a( 1 x 3 + ' x 6 - . . j + b( x - x* + ' x~ - . .] 

\3! 6! /\4! 7! / 

The so-called Riccati's equation, 

= j. hy% = cx n 

ax 

has attracted a lot of attention in the past. Otherwise it is of no particular 
interest here. It is easily reduced to a linear form of the second order. Its 
solution appears as a converging series, finite under certain conditions. 

Forsyth (I.e.) or Johnson (I.e.) must be consulted for fuller details, A de- 
tailed study of the more important series employed in physical mathematics 
follows naturally from this point. These are mentioned in the next section 
along with the titles of special textbooks devoted to their use. 

145. Harmonic Analysis. 

One of the most important equations in physical mathematics, is 

It has practically the same form for problems on the conduction of heat, the 
motion of fluids, the diffusion of salts, the vibrations of elastic solids and 



358 HIGHER MATHEMATICS. 145. 

flexible strings, the theory of potential, electric currents and numberless 
other phenomena, x, y, z are the coordinates of a point in space, t denotes 
the time and V may denote temperature, concentration of a solution, electric 
and magnetic potential, the Newtonian potential due to an attracting mass, 
etc., K is a constant. If the second member is zero, we have Laplace's equation, 
if the second number is equated to irp, where p is a function of x, y, z, the 
result is known as Poisson's equation. 
3 2 F 3F 3 2 F 

+ ~ = ' is Laplace s e( * uatioIL 



3 2 F 3 2 F 3 2 F 

*5-g + 7j~T + 3^2 = 4*v>, is Poisson's equation. 

The first member is written v 2 Fby some writers, A 2 Fby others. The equa- 
tion is often more convenient to use in polar coordinates, viz., 

- v 9 2 F i 3 2 F 2 3F cote BF i 

= 3^2+ -j*' ^p + r'W + ~^~'30 + r 2 ^ 2 ? ' 
where the substitutions are indicated in (11), 48.* 

Any homogeneous algebraic function of x, y, z, which satisfies equation 
(1), is said to be a solid spherical harmonic. These functions are chiefly used 
for finding the potential on the surface of a sphere, due to forces which are 
not circularly symmetrical, f 

Particular solutions of (1) give rise, under special conditions, to the so- 
called surface spherical harmonics, tesseral harmonics and toroidal harmonics. 

The series 



, 



_ 
2 n r(n + 1) 2 2 (n + 1) 2 4 . 2 \(n f 1) (n + 2) 

is called a Cylindrical Harmonic or a BesseVs function of the nth order. The 
symbol J n (x) is used for it. The series is a particular solution of Bessel's 
equation. 

d*y I dy 

+ -+ 



If n = 0, the series is symbolised by J (x) and called a BesseVs function of the 
zeroth wder. These functions are employed in physical mathematics when 
dealing with certain problems connected with equation (1). Another particular 
solution is 



called a Bessel's function of the second kind (of the zeroth order), symbolised 

bv jff o( x )- 

Similarly, the solution of Legendre's equation 

(1 - x^y - 2x-^ + m(m + 1) = 0, 

ax ax 

is the series 

i m ( m + 1 )^2 a. m ( m ~ 2 ) ( m + 1 ) ( m + 3 ) 4 
~2T~ ~TT~ 

* This transformation is described .in the regular textbooks. But possibly the 
reader can do it for himself. 

t A point is said to be circularly symmetrical, when its value is not affected by 
rotating it through an angle about the axis. 



8 145. HOW TO SOLVE DIFFERENTIAL EQUATIONS. 369 

written, for brevity, P m (x). This furnishes the so-called Surface Zonal Har- 
monics, Legendre's coefficients, or Legendrians. Another particular solution, 

x - < m ~ !M m + 2 ) :r s + ( m -!)("*- 3) (m + 2) (m + 4)^ 

3 ! 5 ! 

written Q m (x), gives rise to Surface Zonal Harmonics of the second kind. 
Both series are extensively employed in physical problems connected with 
equation (1). 

The equation 
(x* - 6 2 ) (x* - c 2 ) + x(x> - 6 2 + z 2 - c 2 ) - {m(m + 1) (a; 2 - 6 2 + c*)p\y= 0, 



called Lame's equation, has "series" solution which furnishes LamPs func- 
tions or Ellipsoidal Harmonics, used in special problems connected with the 
ubiquitous equation 



. 

K at 

The so-called hypergeometric or Gauss' series, 
ab a(a + 1)6(6 + 1) 
1 + 1U' + 2lc(c + l) '*+- 
appears as a solution of certain differential equations of the second order, say, 

x(l - x)j + { C -(a + b + l)x}^ - aby = 0, 

(Gauss' equation) , where a, b, c, are constants. 

The application of these series to particular problems constitutes that 
branch of mathematics known as Harmonic Analysis. 

But we are getting beyond the scope of this work ; for more practical 
details, the reader will have to take up some special work such as Byerly's 
Fourier's Series and Splierical Harmonics. Weber and Riernann's Die Par- 
tiellen Differential- Gleichungen der Mathematistfien Physik is the textbook 
for more advanced work. Gray and Mathews have A Treatise on Bessel's 
Functions and tlieir Application to Physics (Macmillan & Co., 1895). 



360 



CHAPTER VIII. 
FOURIER'S THEOREM. 

" Fourier's theorem is not only one of the most beautiful results of 
modern analysis, but may be said to furnish an indispensable 
instrument in the treatment of nearly every recondite question 
in modern physics. To mention only sonorous vibrations, the 
propagation of electric signals along a telegraph wire, and the 
conduction of heat by the earth's crust, as subjects in their 
generality intractable without it, is to give but a feeble idea of 
its importance." THOMSON AND TAIT. 

146. Fourier's Series. 

JUST as a musical note can be resolved into a fundamental note 
and its overtones, so every periodic vibration can be resolved into 
a series of secondary vibrations represented, in mathematical sym- 
bols, by a series of terms arranged, not in a series of ascending 
powers of the independent variable, as in Maclaurin's theorem, 
but in a series of sines and cosines of multiples of this variable. 
Such expansions in a series of trignometrical terms, are of great 
importance in physical problems involving potential, conduction 
of heat, light, sound, electricity and other forms of propagation. 
The series, developed by means of Fourier's theorem, is called 
Fourier's series. 

Any physical property (density, pressure, velocity, etc.) which 
varies periodically with time and whose magnitude or intensity 
can be measured, may be represented by a Fourier's series. This 
means, as we shall soon see, that every vibration can be resolved 
into a series of harmonic vibrations. 

Fourier's theorem determines the law for the expansion of any 
arbitrary function in terms of sines or cosines of multiples of the 
independent variable (x). 



147. FOURIER'S THEOREM. 361 

If f(x) is a periodic function with respect to time, space, tem- 
perature, potential, etc., Fourier's theorem states that 

f(x) = A Q + fljsin x + a.,sin 2# + . . . + ^cos a; -f 6 2 cos 2# + . . . , (1) 
which is known as Fourier's series. A trignometrical function, 
like Fourier's series, for example, passes through all its changes 
and returns to the same value when x is increased by 2?r. See also 
d'Alembert's equation, page 348. 

Assuming this theory to be valid between the limits x = + TT 
and x = - TT, we shall now proceed to find values for the co- 
efficients, A , a v a 2 , . . . , 6j, 6 2 , . . . , which will make the series 
true. 

In view of the fact that the terms of Fourier's series are all 
periodic we may say that Fourier's series is an artificial way of 
representing the propagation or progression of any physical quality 
by a series of waves or vibrations. 

147. Evaluation of the Constants in Fourier's Theorem. 

First, to find a value for the constant A . Multiply equation (1) 
by dx and then integrate each term between the limits x = + IT and 
x = - TT. Every term involving sine or cosine terms vanishes, and 

. dx ; or, A, = + \f(x) . dx, . (2) 



J + '/(x) 



remains. Therefore, when f(x) is known, this integral can be 
integrated.* 

Second, to find a value for the coefficients of the cosine terms, 
say b n , where n may be any number from 1 to n. Equation (1) 
must not only be multiplied by dx, but also by some factor such 
that all the other terms will vanish when the series is integrated 
between the limits + TT, b,,cosnx remains. Such a factor is 
coa nx . dx. In this case, 

J+T 
cos' 2 nx . dx = b n ir, 
-7T 

(page 184), all the other terms involving sines or cosines, when 
integrated between the limits + TT, will be found to vanish. Hence 
the desired value of b n is 

+ *f(x) .cosnx. dx. . . (3) 



* I have omitted details because the reader should find no difficulty in working 
out the results for himself. It is no more than an exercise on preceding work. 



362 HIGHER MATHEMATICS. 148. 

This formula enables any coefficient, b v b 2 , . . . , b n to be obtained. 
If we put n = 0, the coefficient of the first term A assumes the 
form, 

^o = J&o ..... W 

If this value is substituted in (1), we can dispense with (2), and 
write 

f(x) = |6 + fl^sin x + frjcos x + a. 2 sin 2x + 6 2 cos 2x + . . . (5)' 



Finally, to find a value for the coefficients of the sine terms, say 
a n . As before, multiply through with siunxdx and integrate be- 
tween the limits + IT. We thus obtain 



i c+ n 

f(x).m 

7TJ-TT 



sin nx . dx. . . (6) 



There are several graphic methods for evaluating the coefficients of a 
Fourier's series. See Perry, Electrician, Feb. 5, 1892 ; Woodhouse, the same 
journal, April 19, 1901 ; or, best of all, Henrici, Phil. Mag. [5], 38, 110, 1894, 
when the series is used to express the electromotive force of an alternating 
current as a periodic function of the time. 



148. The Development of a Function in a Trignometrical 

Series. 

1. The development of a trignometrical series of sines. Suppose 
it is required to find the value of 

f(x) = x, 
in terms of Fourier's theorem. From (2), (3) and (6), 

1 f + v 1 f + "" 2 

b H = - x . cos nx . dx = ; a n = - x . sin nx . dx = - 

Tj-Tr -n-J-n n 

according as n is odd or even ; 



Hence Fourier's series assumes the form 

x = 2(sin# - | sin2# + Jsin3# - . . . ), . (7) 
which is known as a sine series ; the cosine terms have disappeared 
during the integration. 

By plotting the bracketed terms in (7), we obtain the series 
of curves shown in Fig. 116. Curve 1 has been obtained by 
plotting y = sin a; ; curve 2, by plotting y = | sin Zx ; curve 3, from 
y = J sin Sx. These curves, dotted in the diagram, represent the 
o vertones or harmonics. Curve 4 has been obtained by drawing 



148. 



FOURIER'S THEOREM. 



363 



ordinates equal to the algebraic sum of the ordinates of the pre- 
ceding curves. 




, FIG. 116. Harmonics of the Sine Curve. 

As a general rule, any odd function of x will develop into a series 
of sines only, an even function of x will consist of a series of cosines. 

The general form of the sine development is 

f(x) = a^inx + a 2 sin2# + 3 sin 3x + . . ., . (8) 
where a has the value given in equation (6). 




FIG. 117. Harmonics of the Cosine Curve. 

2. The development of a trignometrical series of cosines. As an 
example, let 



be expanded by Fourier's theorem. Here 
&= - 



_ 4. 
x 2 . cos nx . Ax = + , 



364 HIGHER MATHEMATICS. 

according as n is odd or even. Also, 



Hence, 



148. 



icos2# - icos3# + . . .Y (9) 



By plotting the first three terms enclosed in brackets on the right 
side of (9), we obtain the series of curves shown in Fig. 117 (p. 363). 
The general development of a cosine series is as follows : 

f(x) = %b - b^osx + 6 2 cos2# + . . ., . (10) 

where b has the values assigned in (3). 

EXAMPLES. (1) Develop unity in a series of sines between the limits 
x = ir and x = 0. Here 

M = i. 

a n = -F.fAnnxdx = (1 - cosn-n-) = All - (- 1)}_1 or 

i 



iro 



according as n is odd or even. Hence, from (8), 
1 = -(x + $ sin 3x 



...). . . . (11) 
The first four terms of this series are plotted in Fig. 118 in the usual way. 




FIG. 118. Harmonics of the Sine Development of Unity. 



(2) Show that for x = 
81 " 



(3) Show that 

x sin x = 1 - 4 cos x - I cos 2x + cos 3# - ^ cos 4z + . . . 
between the limits v and 0. 

If* = *then _ + , A + A _... 



(12) 



(13) 






FOURIER'S THEOREM. 



365 



(4) Show x = - (cos x + % cos 3# + ^ 6 cos 5x + . 

Hint. 6 = - I ' 
J ( 

b n = - I x cos nx . da? = ---(cos nir - 1) = 
*J o nV n% 



(14) 



(5) Show that if c is constant, 

c = -(sin x + \ 
tr 

Plot the curve as indicated above. 



(15) 



3. Comparison of the sine and the cosine series. The sine and 
cosine series are both periodic functions of x, with a period of 2?r. 
The above expansions hold good only between the limits x = + TT, 
that is to say, when x is greater than - IT, and less than + IT. 
When x = 0, the series is necessarily zero, whatever be the value 
of the function. 

Now any function can be represented both as a sine and as a 
cosine series. Although the functions and the two series will be 
identical for all 
values of x between 
x = TT and x 0, 
there is a marked 
difference between 
the sine and cosine 
developments of the 
same function. For 
instance, compare FlG ' H9. Diagrammatic Curve of the Cosine Series. 

the graph of x when developed in series of sines and series of 

cosines between the limits x = and x = TT, as shown in (7) and 

(14) above. Plot 

these equations for 

successive values 

of x between + TT, 

etc. In the case of 

the cosine curve 

we get the lines 

shown in Fig. 119. 

By tracing the 

curves correspond- FlG- 120- Diagrammatic Curve of the Sine Series. 

ing to still greater values of TT, we get the dotted lines shown in the 

same figure. For the sine curve we get the lines shown in Fig. 120. 





366 HIGHER MATHEMATICS. $ 149. 

Note the isolated points ( 63) for x = + TT, y = ; x = 3 TT, 
y-O; . . . 

Both these curves coincide with y = x from x = to x = TT, 
but not when x is less than - TT, and greater than + TT. 



149. Extension of Fourier's Series. 

Fourier's series may be extended so as to include all values 
of x between any limits whatever. 

(i.) When the limits are x = + c, x = - c. Let <f>(x) be any 
function in which x is taken between the limits - c and + c. 
Put x = CZ/TT, or z = TTX/C. Hence, 

+(x) = 4>(czlir) = f(z), say. . . (16) 

When x changes from - c to + c, z changes from - TT to + TT, and, 
therefore, for all values of x between - c and + c, the function f(z) 
may be developed as in Fourier's series (5), or 

f(z) = |6 + bjGosz + a^iuz + & 2 cos2 + a 2 siu2z + . . . (17) 

1 /+ TT 1 f + "" 

where, b n = - /(^)cos nz . dz ; a n = ~ f(z)sin nz . dx, . (18) 

Tj-TT J_, 

or, from (16), 

. . . ; (19) 



c 
and from (18), 



(20) 

c J-t- ^ Cj_ e C 

For the sine series, from x = to x = c, 

n ?^ + . 
c 

(22) 



f(x) = a l8 in + a 2 sin + a 3 sin + . (21) 

c c c 



o 
For the cosine series, from x = to # = c, 



l8 in^ +.... (23) 

(24) 

If <f>(x) is a periodic function whose period is equal to c, then, 
(19) is true for all values of x. Hence the rule : Any arbitrary 
function, whose period is T = 2c, can be represented as a series of 
trignometrical functions ivith periods T, \T, \T, . . . 



;i 14'.. FOURIER'S THEOREM. 

EXAMPLES. Prove the following series for values of x from x = to x = c. 

. . (28) 



,8) ^--^'(cos^co^^sin^ +...). . (27) 

Hint. (26) is f(x) = mx developed in a series of sines ; (27) the same function 
developed in a series of cosines. 

(4) If f(x) = x between + c and - c, 

*-*(.!,,- Ida*? +..A . . . (28) 

TT\ C 2 C ) 

(ii.) When the limits are + oo and - GO. Since the above 
formulae are true, whatever be the value of c, the limiting value 
obtained when c becomes infinitely great should be true for all 
values of x. 

In order to prevent mistakes in working, it is usual to employ 
some equivalent sign, say A, for the integrated variable x. Hence 
in place of equations (20), we may write 



b " " ^ (A) COS dX ; a " ~ ^ (X) Sin A ' (29) 
Substitute these values in (23), 

f(x) = *{|f * /(^A + J + >(A) cos ^ cos ^ 

+ f + /(A)sin sin-^A + . . A ; 

J -c- G C 

lf + %/x \jxfl wX ir . TrA . irX \ 

= - t(X)dX\ - + cos cos + sin sin 4. 

cj _; (2 c c ' c c )' 



cos - - 



. . . + ^cos^A - x) + ^cos^(A - a;) 

C C C C 



since cos = 1; for the other trignometrical substitutions, see 
page 499. As c is increased indefinitely, the limiting value of the 

f+GO 
cos a(A - x)da, where a = nir/c, n being 
-00 

any integer. Hence, the limiting form oif(x) is 



f + 

J -00 



(A - x)da, . . (30) 



368 HIGHER MATHEMATICS. 150. 

for all values of x. The double integral in (30) is known as 
Fourier's integral. 

150. Different Forms of Fourier's Integral. 

Fourier's integral may be written in different equivalent forms. 

From 80, 

J + oo f o p x 

cos xdx = \ cos xdx + \ cos xdx ; 
- 00 J - X JO 

rro ro 

cos xdx = I cos ( - x)d( - x) = - I cos xdx ; 
- oo J oo J oo 

J + oo r*> 

cos xdx = 2 1 cos xdx. 
-oo Jo 

Hence, we may write in place of (30), 

/(*) = ~ J + _ "/(A)dxcos a(A - x)da, . . (31) 

where the integration limits in (31) are independent of a and A, and 
therefore the integration can be performed in any order. 
Let /(a?) = -/( - x). Then, 

f(x)= fMd\\ X cosa(\-x)da = ~ da\ /(A)cosa(A-z)dA;, 

^ J - <* Jo ^0 J-oo 

I f + co TO f< 

da\ /(A) COS a(A - x)d\ + f(\) COS a(\ - x}d\ ; 

''"J-ooJ-oo Jo 

I f + 00 fO pGO 

^a /( - X) cos a( - A - x)d( - A) + /(X) cos a(X - x)d\ ;. 

^J Joo Jo 

1 r a r f ^1 r 00 

^a] - /(A) cos a(A + x)d\ } + /(A) cos a(A - x}dX ; 
^Jo I Jo Jo 

If* f* 

da\ /"(A){cosa(A - x) - cosa(A + x)\d\ ; 
''"Jo Jo* 

2/-00 |.00 

I da. \ /(A) sin aA . sin ax . d\ ; 

77 Jo Jo 

2 f r 

= - /(A)<ZA sin aA . sin ax . da, (32) 

77 Jo Jo 

which is true for all odd functions of f(x) and for all positive values 
of x in any function. 

Let f(x) = /( - x), we can then reduce (31) in the same way to 

2 poo -oo 

f(x) = - fMd\\ cos aA . cos ax . da, . . (33) 

^Jo Jo 

which is true for all values of x, when f(x) is an even function of 

#, and for all positive values of x in any function. For the trigno- 
metrical reductions, see page 499. 



$ 151. FOURIER'S THEOREM. :m 

Although the integrals of Fourier's series are obtained by inte- 
grating the series term by term, it does not follow that the series 
can be obtained by differentiating the integrated series term by 
term, for while differentiation makes a series less convergent, 
integration makes it more convergent. In other words, a con- 
verging series may become divergent on differentiation. This 
raises another question, the convergency of Fourier's series. 

151. The Convergency of Fourier's Series. 

In the preceding developments it has been assumed : 

1. That the trignometrical series is uniformly convergent. 

2. That the series is really equal to f(x). 

Most elaborate investigations have been made to find if these 
assumptions can be justified. The result has been to prove that 
the above developments are valid in every case when the function 
is single-valued * and finite between the limits + TT,* and has only 
a finite number of maximum or minimum values,* between the 
limits x = TT. 

The curve y = f(x) need not follow the same law throughout 
its whole length, but may be made up of several entirely different 
curves. A complete representation of a periodic function for all 

* The terms marked with the asterisk may, perhaps, need definitions. According 
to 78, the integral 

r <**-. RT 2 
JJ^ UJ-, 

represents the area included between the curve y I/a: 2 , the #-axis, and the ordinates 
drawn from x - 1 and x - 1. Plot the curve and you will find that this result is 
erroneous. The curve sweeps through infinity, whatever that may mean, as x passes 
from -f 1 to 1 (see 52). The method of integration is, therefore, unreliable when the 
function to be integrated becomes infinite or otherwise discontinuous at or between the 
limits of integration. Consequently, it is necessary to examine certain functions in 
order to make sure that they are finite and continuous between the given limits, or 
that the functions either continually increase or decrease, or alternately increase 
(maxima) and decrease (minima) a finite number of times. This subject is discussed in 
the opening chapters of Riemann and Weber's The Partial Differential Equations of 
Mathematical Physics (German F. Vieweg & Sons, Braunschweig, 1900-1901), to 
which the student must refer if he intends to go exhaustively into these questions. 

Single-valued and multiple-valued functions have been defined on page 275. 
y = tan - *x is a multiple-valued function, because the ordinates corresponding to 
the same value of x differ by multiples of TT. Verify this by plotting. Obviously, if 
x = a and x b are the limits of integration of a multiple-valued function, we must 
make sure that the ordinates x - a and x = b belong to the same branch of the curve 

y =AX)- 

AA 



370 HIGHKIl MATHEMATICS. 152. 

values of x would provide for developing each term as a periodic 
series, each of which would itself be a periodic function, and so on. 

A discussion of the conditions of convergency of Fourier's series must be 
omitted from this chapter. Byerly's An Elementary Treatise on Fourier's 
Series, etc., is one of the best practical works on the use of Fourier's integrals 
in mathematical physics. Fourier's ancient Theorie analytique de la Chaleur 
of 1822 is perhaps as modern as any other work on this subject.* See also 
Williams, Phil Mag. [5], 42, 125, 189G ; Lord Kelvin's Collected Papers; and 
Weber- Riemann's work (I.e.). 

152. The Superposition of Particular Solutions to Satisfy 
given Conditions. 

The following remarks will amplify what has already been said 
in connection with this important principle. 

The reader knows that ordinary and partial differential equa- 
tions differ in this respect : While ordinary differential equations 
have only a finite number of independent particular integrals, partial 
differential equations have an infinite number of such integrals. 

To show that a value of V, in Laplace's equation, can be ob- 
tained to satisfy Fourier's integral (31). Suppose that a value of 
V is required in the equation 



.* (34) 

such that when y = CD, V = 0, and when y 0, V = f(x). First 
assume that 

V = e*y + & x , 

is a solution, when a and ft are constant. Substitute in (34) and 
divide by ey + fa. 

... ^ + p 2 = 0, 

if this condition holds, the above value of V is a solution of (34). 
Hence 7 = e a v La - r , are solutions of (34), therefore also e a ve tax and 
e a.y e uuc are so i u tions. Add and divide by 2, or subtract and 
divide by 2, 112, (3) and (4), thus 

V = ey cos ax ; and V = e-y sincur, . . (35) 
are solutions of (34). Multiply the first by cos aA. and the second 
of (35) by sin aA. The results still satisfy (34). Add, (22), page 499, 
and 

- x) 



* Freeman's translation can sometimes be obtained from the second-hand book- 
sellers. 



g 152. FOURIER'S THEOREM. 1 371 

satisfies (34). Multiply by f(\)d\ and the result is still a solution 
of (34) 

e ~ *vf(X) . cos a(A - x)d\. 

Multiply by I/TT and find the limits when a has different values 
between and oo. Hence 



= - f 
^"Jf 



e~ vf(\) cos a(\ - x)d\, (36) 



satisfies the required conditions. According as f(x) is an odd or 
an even function, the right-hand side of (36) reduces to 

2f* f* 

I da. I e - a?/ /TA) cos ax cos aA. . d\ ; 

^Jo Jo 

2f x f 

or to I da I e ~ a vf(\) sin ax sin aX . d\. 

''"Jo Jo 

EXAMPLES. (1) A large iron plate IT cm. thick and at a uniform temperature 
of 100 is suddenly placed in a bath at zero temperature for 10 seconds. Re- 
quired the temperature of the middle of the plate at the end of 10 seconds, 
supposing that the diffusivity K of the plate is 0-2 C.G.S. units, and that the 
surfaces of the plate are kept at zero temperature the whole time. 

If heat flows perpendicularly to the two faces of the plate, any plane 
parallel to these faces will have the same temperature. Thus V depends on 
one space coordinate, equation (1), 145, reduces to 



The conditions to be satisfied by the solution are that V = 100, when = 0; 
T" = 0, when x = ; F = 0, when x = ir. 

First, to get particular solutions. Assume F = e* + & 8 is a solution when 
a and j8 are constants. Substitute in (37) and divide by e** + & . Hence 
=" a 2 , provided F = e** + & 9 , is a solution of (37). This is true whatever 
be the value of o, hence F = e"* + & Q is also a solution of (37) for all values of 

a. Put a = tn, where t = V^T. Then F = e ~ w\ l n*, and V=e~ *^ e ~ ^ 
are solutions of (37). 

.-. r=$ e - &( e w + e - ^ = e ~ *& cos fix, . . (38) 
is a solution of (Si). Similarly, 

F = e ~ Kl ^ e sin /tx, ..... (39) 

is a solution of (37), whatever be the value of p. By assigning particular 
values to /*, we shall get particular solutions of (37). Cf. footnote, p. 306. 

Second, to combine tJiese particular solutions so as to get a solution of (37) 
to satisfy the three given conditions, we must observe that (39) is zero when 
x = 0, for all values of /*, and that (39) is also zero when x = ir if /* is an 
integral number. If, therefore, we put F equal to a sum of terms of the 

form Ae ~ * M 6 sin nx, say, 

F = a^e ~ * 9 sin x + a# ~ *" e sin 2x + a 3 e ~ 9 e sin Sx + ..... (40) 



372 HIGHER MATHEMATICS. 152. 

to n terms, this solution will satisfy the second and third of the above 
conditions, because sin IT = = sin 0. When = 0, (40) reduces to 

V = Oj sin x + 2 sm % x + a s s i n 3x + . . . . . (41) 
But for all values of x between and TT, (11), 

1 = -(sin x + sin 3x + | sin 5x + ...),. . . (42) 

if V = 100, we must substitute the coefficients of this series multiplied by 
100, for On 03, a 3 , . . . in (40), to get a solution satisfying all the required 
conditions. Note 2 , 4 , . . ., in (42), are zero. We thus obtain 

V = ^-(e ~ * e sin x + \e ~ 9 * sin 3x + . . .) . . (43) 

This is the solution required. 

To introduce the numerical data. When x = TT, 6 = 10, /c = 0*2. Hence 
use a table of logarithms. The result is accurate to the tenth of a degree if 
all terms of the series other than the first be suppressed. Hence use 



for the numerical calculation. Use Table XXII. Ansr. 17-2 C. 

Byerly (I.e.) has a splendid collection of problems of this nature. I have 
arranged a set of greater interest to the chemist at the end of this chapter. 

(2) If the plate is c centimetres instead of TT centimetres thick, use the 
development 

^SC-T + i*?* -'> 

from x = 1 to x = c. 

(3) An infinitely large solid with one plane face has a uniform tempera- 
ture f(x). It the plane face is kept at zero temperature, what is the tempera- 
ture of a point in the solid x feet from the plane face at the end of t years ? 

Let the origin of the coordinate axes be in the plane face. We have to 
solve equation (37) subject to the conditions 

V= when x = 0; V=f(x) when t = 0. 

Proceed according to the above methods for (38), (39), and (36). We thus 
obtain 

1 /*oo ^-j-oo 

V = - da I e~ Ka2e /(A)cos a(A - x).d\; 

*J 9 J -oo 

since positive values of x are wanted we can write 

V = -I da I e~ ca */(X) sin ax . sin a\.d\, 

*J o J o 
as above. Hence from (26), page 499, 



- X) - GOSa(A + x)}da. 
-a) 2 



is the required solution. 

This last integration needs amplification. To illustrate the method, let 

u = j e ~ a ' 2x ' 2 cos bx . dx. 
Laplace (1810) first evaluated the integral on the right by the following 



$ 1.VJ. FOURIER'S THEOREM. 373 

ingenious device which has been termed integration by differentiation. Dif- 
ferentiate the given equation and 



provided 6 is independent of x. Now integrate the right member by parts in 
the usual way (page 168), 

du b du b ,, 

''db = - W U ' r ^ = * W db - 
Integrate, and 

A2 &" 

log it = - -r-z + C ; or u = Ce ~ 4a*- 
4& 

To evaluate C, put 6 = 0, whence 



2a ' 
as in (12), page 191. Therefore 

r _ > > VTT 

% = / e cos bx .dx = -^r e 

Jo 2 

Returning, after this digression, to the original problem, it follows that 

if we write for brevity = (A. - x)/2 *Jttt, 

\ir \J 7= J + 7^ J 

Z\tct 2\ft 

If the initial temperature is constant, say = F , 



from (4), page 185, and page 368. 

For numerical computation it is necessary to expand the last integral in 
series as described on page 270. Therefore 



F f x x 3 \ 

N/W 12 x/irf 3 . (2 x'jtf) 3 + ' ' ' J ' 



If 100 million years ago the earth was a molten mass at 7,000 F., and, 
ever since, the surface had been kept at a constant temperature F., what 
would be the temperature one mile below the surface at the present time, 
taking Lord Kelvin's value K = 400 ? Ansr. 104 F. (nearly). Hints. F = 7,000 ; 
x = 5,280 ft. ; t = 100,000,000 years. 

.2 x 7,000/ 5280 \ _ 

" \2 x 20 x 10,0007 



Lord Kelvin, "On the Secular Cooling of the Earth" (Thomson and Tait's 
Treatise on Natural Philosophy, 1, 711, 1867), has compared the observed 
values of the underground temperature increments, dVjdx, with those deduced 
by assigning the most probable values to the terms in the above expressions. 
The close agreement (Calculated : 1 increment per 3 \ ft. descent. Observed : 

* NOTE, oo + a finite quantity = x . It is not difficult to show either by graphic 
construction or by integration that 

.6 J,-x J> J> + X 

I (t - x)t.dt= z(z + x)dz ; / (t + x)t . dt = / z(z - x)dz. 
J a J a-x J a J a + x 



374 



HIGHER MATHEMATICS. 



153. 



1 increment per ^ ft. descent) leads him to the belief that the data are 
nearly correct. He extends the calculation in an obvious way and concludes : 
" I think we may with much probability say that the consolidation cannot 
have taken place less than 20,000,000 years ago, or we should have more 
underground heat than we really have, nor more than 400,000,000 years ago, 
or we should not have so much as the least observed underground increment 
of temperature ". Vide Heaviside's Electromagnetic Theory, 2, 12, 1899. 



153. Fourier's Linear Diffusion Law. 

Let AB be any plane surface in a metal rod of unit sectional area 
(Fig. 121). Let this surface, at any instant of time, have a uni- 
form temperature (equithermal surface), and let the temperature 
on the left side of the plane AB be higher than that on the right. 
In consequence, heat will flow from the hot to the cold side, in 
the direction of the arrow, across the surface AB. 

Fourier assumes, 

1. The direction of the flow is perpendicular to the surface AB; 

2. The rate of flow of heat across any given surface, is pro- 
portional to the difference of temperature on the two sides of the 
plate. 

Now let the rate of flow be uniform, the temperature of the 
plane AB, 0. The rate of rise of temperature at any point in the 



hot 




B 



plane AB, is dO/ds (this ratio 
measures the so-called tem- 
perature gradient). The 
amount of heat which flows, 
per second, from the hot to 
the cooler end of the rod, is 
- c.dO/ds, where c is a con- 
stant denoting the heat that 
flows, per second, through 
(Why the 





FIG. 121. 

unit area, when the temperature gradient is unity, 
negative sign?) 

Consider now the value of - c . d6/ds at another point in the 
plane CD, distant 8 s from AB ; this distance is to be taken so small, 
that the temperature gradient may be taken as constant. The 



temperature at the point s + 



(, will be (6 - ^os), since -dO/ds is 

as ' 



the rate of rise of temperature along the bar, and this, multiplied 
by 8s, denotes the rise of temperature as heat passes from the 



S 153. FOURIEB'8 TIIKoiiKM. :;::. 

point s to s + Ss. Hence the amount of heat flowing through the 
small section ABCD will be 

d ( dO 

~ C J-( e - T- 

ds\ ds 

will denote the difference between the amount of heat which flows 
in at one face and out at the other. This expression, therefore, 
denotes the amount of heat which is added to the space ABCD 
every second. If a- denotes the thermal capacity of unit volume, 
the thermal capacity of the portion ABCD is (1 x 8s)cr. Hence 

df) 
the rate of rise of temperature is <r 8$. Therefore, 

(it 

d 2 o. do. 

. c^> s ^T*. 
ds' 2 dt 

Put o/<r = K ; this equation may then bs written, 

d*0 1 dO , 44 v 

d* 2 = - K 'dt' 
where K is the diffusivity of the substance.* 

Equation (44) represents Fourier's law of diffusion. It covers 
all possible cases of diffusion where the substances concerned are 
in the same condition at all points in any plane parallel to a given 
plane. It is written 

*T.i.?i (45) 

^X 2 K t> 

Lord Kelvin calls V the quality of the substance at the time t, 
at a distance x from a fixed plane of reference. The differential 
equation (45), therefore, shows that the rate of increase of quality 
per unit time, is equal to the product of the diffusivity and the 
rate of increase of quality per unit of space of quality. The quality 
depends on the subject of the diffusion. For example, it may 
denote one of the three components of the velocity of the motion 
of a viscous fluid, the density or strength of an electric current per 
unit area perpendicular to the direction of flow, temperature, the 
potential at any point in an isolated conductor, or the quantity of 
salt diffusing in a given solution. 

Ohm's law is but a special case of Fourier's linear diffusion 
law. Fick's law of diffusion is another. The transmission of 
telephonic messages through a cable and indeed any phenomenon 
of linear propagation is included in this law of Fourier. 

* This equation is obtained from (1), 145, by remembering that we are dealing 
only with linear flow, in one direction ; the y and z terms do not, therefore, come in at 
all. (44) is a special case of that more general equation. 



376 HIGHER MATHEMATICS. $ 154. 

EXAMPLES. (1) Show by actual differentiation that (45) is satisfied by 
F= e-^sintft - ax), where o, $ are two constants such that 0/2a 2 = K. Hint, 
first show that 3 2 7/9o: 2 = e - **2a?cos(pt - ax), then that 
'dVf'dt = e ~ **p cos(0< - ax), etc. 

(2) Show that (45) is satisfied by making 

V = e - <2a 2 cos(j8< - ax), 

or V= A + A^e ~ a i r siu(^t - Ul x + ei ) + A#~ ^s'm(ft 2 t - a^x + e 2 ) + . . ., 
where A , A lt . . ., e 1? 2 , . . ., are constants. See page 350. 

(3) Deduce Fick's law of diffusion, similar in form to (45), for a salt solu- 
tion in a vertical vessel of uniform sectional area, the solution being more 
concentrated in the lower part of the vessel. Assume (1) the rate of diffusion 
(quantity of salt passing through unit sectional area in unit time) is pro- 
portional to the difference in the concentration on each side of a given 
horizontal plane, (2) the substance diffuses in a vertical direction. Hint, 
follow the discussion preceding (44), and make the proper changes. 

If V denotes the concentration of the solution in any plane x, at any time 
t, Fick's law is written, 



where K depends on the nature of the diffusing substance. 

154. The Solution of Fick's Equation in terms of a 
Fourier's Series. 

The experimental basis of the following discussion will be 
found in a memoir by Simmler and Wild in Poggendorffs Annalen 
for 1857 (100, 217, 1857) : Fill a small cylindrical tube of unit 
sectional area, and height x, with a solution of salt. Let the tube 
and contents be submerged in a vessel containing a great quantity 
of water so that the open end of the cylindrical vessel, containing 
the salt solution, dips just beneath the surface of the water. Salt 
solution passes out of the diffusion vessel and sinks towards the 
bottom of the larger vessel. The upper brim of the diffusion 
vessel, therefore, is assumed to be always in contact with pure 
water. Let h denote the height of the liquid in the diffusion tube, 
reckoned from the bottom. 

(i.) To find the concentration (V) of the dissolved substance in 
different parts (x) of the diffusion vessel after the elapse of any 
stated interval of time (t). 

This is equivalent to finding a solution of Fick's equation, (46), 
of the preceding section, which will satisfy the conditions under 
which the experiment is conducted. These so-called " limiting 
conditions " are : 



^ I.-.4. FOURIER'S THEOREM. 377 

.When x = h, V = ; . . . . (1) 

when x = 0, D7/fo - 0; ... (2) 

when t = 0, V = 7 ..... (3) 

The reader must be quite clear about this before going any further. 
What do 7, a; and t mean ? F evidently represents the concen- 
tration of the solution at the beginning of the experiment ; at the 
top of the diffusion vessel, obviously x = h, and 7 is zero, because 
there the water is pure ; the second condition means that at the 
bottom of the diffusion vessel, the concentration may be assumed 
to be constant during the experiment. 

First, deduce particular solutions exactly as in the first example 
of the preceding section. Thus 

7 = ae-'AsosyuuE, ... (4) 

and V = be-^smpXj ... (5) 

are particular solutions, a and b simply denote arbitrary constants. 

Differentiate (4) and we get 



Now when x 0, sin/xic vanishes, therefore, when x = 0, condi- 
tion (2) is satisfied. But, in order that (4) may satisfy the first 
condition, we must have 

cos fih = 0, when x = h. 

But COS \TT = COS f TT = . . . = COS (2tt - !)TT = 0, 

where TT = 180 and n is any integer from 1 to oo. Hence, we 
must have 



' W W Ml ' 2k ' 

in order that cos /uA may vanish. 

Substitute these values of //, successively in (4) and add the 
results together, we thus obtain 






7= Oja'aM cosr + a. 2 e C os -- + ... to inf., (6) 

which satisfies two of the required conditions, namely (1) and (2). 

We must now determine the coefficients a lt a 2 , ... in (6), in 
order that the third condition may be satisfied by the particular 
solution (4), or rather (6). This is done by allowing for the initial 
conditions, when t = 0, in the usual way. When t = 0, 7 = 7 . 
Therefore, from (6), 

TTX 3-n-x 

7 = a x cos + a., cos - + . . ., . . (7) 



378 HIGHER MATHEMATICS. $ 154. 

is true for all values of x between and h. Hence, as on page 
2F A 



2F f* 

; a * = irJ cos 



These results have been obtained by equating each term of (7) to 
zero, and integrating between the limits and h. 

Substituting these values of a Q , a lt . . . in (6), we get a solu- 
tion satisfying the limiting conditions of the experiment. If desired, 
we can write the resulting series in the compact form, 



_ 
1 5TTL' cos Tf6 (8) 

where the summation sign between the limits n = GO and n = 1 
means that n is to be given every positive integral value 1, 2, 3. ... 
to infinity, and all the results added together. 

EXAMPLE. If we reckon h from the top of the diffusion vessel, show that 
we must use (5) exactly as we have just employed (4). In this case " cos " 
(8) becomes " sin ". 

NOTE. If the limits in (8) are o and oo , write " 2n + 1 " for " 2n - 1 ". 

(ii.) To find the quantity of salt (Q) which diffuses through any 
horizontal section in a given time (T). 

Differentiate (6) with respect to x, multiply the result through 

dF 

with Kdt, so as to obtain - K^dt. If x represents the height of 

any horizontal section, - nq-^-dt, will represent the quantity of 

salt which passes through this horizontal plane in the time dt. 
q represents the area of that section (example (3), page 374). 
Let q = I. 

f^ 7 ^ fW -()'" ** 

* g = - "ts* = ]5( v sm M + - 

Integrate between the limits and T. The result represents the 
quantity of salt which passes through any horizontal plane (x) of 
the diffusion vessel in the time T, or, 



(iii.) To find the quantity of salt (QJ which passes from the 
diffusion vessel in any given time (T). 

Substitute h = x in (9). The sine of each of the angles 
JTT, |TT, . . ., (2w - 1), is equal to unity. Therefore, 

. (10) 



$154. FOURIER'S THEOREM. 37'. 

(iv.) To find the value of K, the coefficient of diffusion. 

Since the members of series (10) converge very rapidly, we 
may neglect the higher terms of the series. Arrange the experi- 
ment so that measurements are made when x = h, \h, \h, . . ., 
in this way, sin 7rx/2h, ... in (9) become equal to unity. We 
thus get a series resembling (10). Substitute for the coefficient 
and we obtain, by a suitable transposition of terms, 



# 



(v.) To find the quantity of salt (Q. 2 ) which remains in the 
diffusion vessel after the elapse of a given time (T). 

The quantity of salt in the solution at the beginning of the 
experiment may be represented by the symbol Q Q . Q may be 
determined by putting t = in (9) and eliminating sin -n-x/Qh, . . . 
as indicated in (iv.). 



Co = (a>i ~ \<*>- 
and Q, = Q -Q l ; 



(vi.) To find the concentration of the dissolved substance in 
different parts of the diffusion vessel when the stationary state is 
reached. 

After the elapse of a sufficient length of time, a state of equili- 
brium is reached and the concentration of the substance in different 
parts of the vessel remains stationary. In this case, 

Wfit = 0; W/Da? = 0. 
Integrate the latter, we get 

V = ax + b, . . . . (13) 

where a and b are constants to be determined from the experi- 
mental data, as described in 106. See Ostwald's Solutions, 
Chapter vi. (Longmans, Green & Co., 1891), for experimental work. 

The chief difficulty in the application of Fourier's theorem to 
diffusion experiments is to make the series satisfy the limiting 



380 HIGHER MATHEMATICS. 154. 

conditions. The following examples will serve to show how 
Fourier's series is to be employed in practical work. For the 
experimental details, the original memoirs must be consulted. 

EXAMPLES. It will be found convenient to refer to the following alter- 
native way of writing Fourier's series : 

) . d, + |2f; ". ^ cos ^,M .<**, . (U) 

true for any value of x between and c (see pages 365 and 366). 

(1) Find an expression equal to v when x lies between and a, and equal 
to zero, when x lies between a and b. Here /(A) = v, from A = to A = a, and 

f(\) = 0, from A = a to A = b ; c = b ; cos ' /(A) . d\, becomes 
or, sin ' . Hence the required expression is, 



when x = a, this expression reduces to %v. 

(2) Pick's diffusion experiments (Pogg. Ann., 94, 59, 1855 ; translated in 
the Phil. Mag., July, 1855). When deducing Fick's equation, if the area of 
the diffusion vessel is some function of its height x, show that Fick's equation 
assumes the form 



where q denotes the area of the diffusion vessel at a distance x in the direction 
of the diffusion. 

Before this formula can be of any practical use, the equation to the curve 
described by the walls of the vessel must be known. For a conical vessel, 
q = 7rw 2 .r 2 , where the apex of the cone is at the origin of the coordinate axes, 
m is the tangent of half the angle included between the two slant sides of the 
vessel. Fick has made a series of crude experiments on the steady state in a 
conical vessel with a circular base (funnel-shaped). Hence show that, 

gJVllT-o^-.r-c. + fli . . . as) 

Bar 2 x ^)x x 

The integration constants C l and C 2 are to be evaluated by means of the 
experimental data, 106. 

(3) Graham's diffusion experiments (Phil. Trans., 151, 183, 1861). A cylin- 
drical vessel 152mm. high, and 87 mm. in diameter, contained 0'7 litre of 
water. Below this was placed 01 litre of a salt solution. The fluid column 
was then 127 mm. high. After the elapse of a certain time, successive portions 
of 100 cc., or | of the total volume of the fluid, were removed and the quantity 
of salt determined in each layer. 

The limiting conditions are : At the end of a certain time t, for x = and 
x = H , 'dVj'dx = 0. (Why?) Note that x is here reckoned from the top of 
the liquid. H denotes the total height of the liquid column. Let h denote 
the height of the salt solution at the beginning of the experiment, V Q its con- 
centration, .-. h = iff; /(A), in (14), = V from x = to x = h and /(A) = V= 0, 
from x = h to x = H, when t = 0. 



g 154. FOURIER'S THEOREM. :WL 

To adapt these results to Fourier's solution of Fick's equation, first show 
that 

V= (acosfjLX + bBmf*x)e-^ Kt , .... (17) 

is a particular integral of Fick's equation, a, 6, are constants to be determined 
from the conditions of the experiment. Differentiate (17) with respect to x 
and we get 

'dV/'dx = ( - fM sin /JLX + /u&cos/i.r)^"'* 2 **. . . (18) 

In the layer x = 0, 'dVf'dx = 0, whatever the value of t, because no salt 
goes out from and no salt enters the solution at this point. The concentra- 
tion V must at all times satisfy Fick's elementary law, at all points between 
x = and x = H . When x 0, cos* = 1, but sin a; = 0, therefore, from (18), 

rtsin/ij; - bcosfj.x = 0, 
b must be zero, and, since sin ir = 0, /t must be so chosen that 

p.H = nit ; or, p. = nw/H, 
where n has any value 0, 1, 2, 3, . . . 

Add up all these particular integrals for the general equation 



e 

where the constant a has still to be determined from the initial conditions. 
For t = 0, 

V = V n = "a n cos(^\ = V, from x = 0, to x = h ; 

H ) 

1 \ = 0, from x = h, to x = H. 

H ) 

Since /(A) = 7 , in (14), when n = 0, 



In the same way it can be shown that 



= . 

ir n H ' 

Taking all these conditions into account, the general solution appears in the 
form, 



which is a standard equation for this kind of work. In Graham's experi- 
ments, h = H. Hence the concentration V in any plane x of the diffusion 
vessel, is obtained from the infinite series : 

7 = Zo + 2Io v = - 1 8in n . C o S !^ . e - 'MM*. . (21) 

8 v i * H 

As indicated in Chapter V., an infinite series is practically useful only 
when the series converges rapidly, and the higher terms have so small an 
influence on the result that all but the first terms may be neglected. This is 
often effected by measuring the concentration at different levels x, so related 
to H that cos(nirx/H) reduces to unity; also by making t very great, the second 
and higher terms become vanishingly small. See Weber's experiments below. 



382 HIGHER MATHEMATICS. 154. 

The quantity of salt Q r in the rth layer, is given by the integral of Vdx, 
between the limits x = (? - l)H and x = %rH, or, 



. 

where \V Q H multiplied by the cross section of the vessel (here supposed unity) 
denotes the quantity of salt present in the diffusion vessel. 

Unfortunately, a large number of Graham's experiments are not adapted 
for numerical discussion, because the shape of his diffusion vessels, even if 
known, would give very awkward equations. A simple modification in ex- 
perimental details, will often save an enormous amount of labour in the 
mathematical work. 

(4) Stefan's diffusion experiments (Wien.-Akad. Ber., 79, ii., 161, 1879). If 
Qo denotes the quantity of salt present in the diffusion vessel of Graham, when 
t is very great, show, preceding example, that 

Qi+ Q4+Q 5 +Qs = &V<>H = $Q< (23) 
where q denotes the area of a cross section of the vessel. 

When deducing (23) from (22), it is most instructive to compile a table of 
values of the factor cos(2r - l)>nr/16, for values of r from r = 1 to r = 8, and 
from n 1 to n = 4. Then show that for n = 1, 2, 3, the sums, for values of 
r = 1, 4, 5, 8, mutually cancel each other, and that the value of t in the higher 
terms makes them negligibly small. Here is the table : 





n = I 


n= 2. 


n = 3. 


n = 4. 


? 


= 1 


+ COS T VT . 


+ cos^ir 


+ cos fVr 


+ cos |T 


r 


= 2 


+ cos -jSg-ir 


+ COS fir 


- cos T V 


- COS |ir 


r 


Q 


+ COS Y\7T 


- cos |TT 


- cos T Vr 


- COS^ir 


r 


= 4 


+ cos far 


- COS^TT 


- COS Y^TT 


+ COS |TT 


r 


= 5 


- COS T 7 ^T 


- COS |TT 


+ cos ^ir 


+ COS \ir 


r 


= 6 


- COS T \T 


COS ^7T 


+ cos T Vr 


COS |ir 


r 


= 7 


- cos -for 


+ cos fir 


+ COS-^ir 


- cos |TT 


r 


= 8 


- COS^T 


+ COS^TT 


- COS T \ir 


+ COS |ir 



Equation (23) has been verified by Stefan. He made Q = 10 grms. of 
salt. The successive layers, from r = 1, to r = 8, contained, after 7 days, 
3-294, 2-844, 1-907, 1-100, 0-529, 0-215, 0-079 and 0-030 grms. of salt. The ex- 
perimental details were similar to those of Graham already described. Show 
that theory requires %Q (} = 5, and that the above numbers furnish %Q Q = 4-953 
a very good agreement. 

(5) Weber's diffusion experiments (Wied. Ann., 7, 469, 536, 1879). A con- 
centrated solution of zinc sulphate (0-25 to 0-35 grm. per c.c. of solution) was 
placed in a cylindrical vessel on the bottom of which was fixed a round smooth 
amalgamated zinc disc (about 11 cm. diam.). A more dilute solution (0-15 to 
0-20 grm. per c.c.) was poured over the concentrated solution, and another 
amalgamated zinc plate was placed just beneath the surface of the upper 
layer of liquid. It is known that if V lt F 2 denote the respective concentra- 
tions of the lower and upper layers of liquid, the difference of potential E, due 
to these differences of concentrations, is given by the expression 

E = A(V 9 - Fj){l + B(V 2 + VJ}, . . . (24) 



3 !f,4. FOURIER'S THEOREM. 383 

where A and B are known constants, B being very small in comparison with 
A. This difference of potential or electromotive force, can be employed to 
determine the difference in the concentrations of the two solutions about the 
zinc electrodes. 

To adapt these conditions to Fick's equation, let ^ be the height of the 
lower, h. 2 of tne upper solution, therefore, 1^ + h 2 = H. The limiting con- 
ditions to be satisfied for all values of t, are 3F/9 = 0, when x = 0, and 
^ Vfdx = 0, when x = H. The initial conditions when t = 0, are V = F 2 , for 
all values of x between x = h and x = H. From this proceed exactly as in 
example (3), and show that 






_ 

H IT n H 

and the general solution 

y.FA+KA _ Ok-I2f -n.to^o.^.**. (25) 

H TT n = \ n H H 

This equation only applies to the variable concentrations of the boundary 
layers x = and x = H. It is necessary to adapt it to equation (24). Let 
the layers x = and x = H, have the variable concentrations V and V" 
respectively. 




In actual work, H was made very small. After the lapse of one day 
(t = 1), the terms Jsin kirh^H, etc., and i sin 5-irhJH, etc., were less than j^. 
Hence all terms beyond these are outside the range of experiment, and may, 
therefore, be neglected. Now h was made as nearly as possible equal to \H, 
in order that the term sin STrhJH, etc., might vanish. Hence, 



V" - V = 2 ~ i sin * e - 

IT 8 

2(F 2 - 7J 

Now substitute these values of V" - V and V" + V in (24), observing that 
F 2 , V lf 7^, 7i2, sin 7r/3, sin 2ir/3 and H, are all constants. The difference of 
potential E, between the two electrodes, due to the difference of concentration 
between the two boundary layers V and F", is 

E = A l e-** t l H *-B l e-^* t l H2 , . . . (26) 

where A l and B l are constant. Since B is very small in comparison with A, 
the expression reduces to 

E = A l e*' tia *, ..... (27) 
in a very short time. 



384 HIGHER MATHEMATICS. ;< 1.V4. 

This equation was used by Weber for testing the accuracy of Fick's law. 
The values of the constant, ir 2 */// 2 , after the elapse of 4, 5, 6, 7, 8, 9, 10 days 
were respectively -2032, -2066, -2045, -2027, -2027, '2049, -2049. A very satisfac- 
tory result. 

(6) A gas A, obeying Dalton's law of partial pressures, diffuses into an- 
other gas, show that the partial pressure p l of the gas A, at a distance x, in 
the time t, is 



(7) Loschmidt' 's diffusion experiments (Wien.-Akad. Ber. t 61, 367, 1870 ; 
62, 468, 1870). Loschmidt arranged two cylindrical tubes vertically, so that 
communication could be established between them by a sliding metal plate. 
Each tube was 48*75 cm. high and 2-6 cm. in diameter and closed at one end. 
The two tubes were then filled with different gases and placed in communica- 
tion for a certain time t. The mixture in each tube was then analysed. 

Let a = 97'5 cm. It is required to solve equation (28) so that when t = 0, 
Pi = Po> from x = a to x = i 5 Pi = 0. fro na x = Ja to x = a ; 3ft/3ar = 0, when 
x = and x = a, for all values of t. Note, ^ denotes the original pressure of 
the gas. Hence show that 

ft = 2j + 2" = "Ism f cos ^ - * At/a.. . . (29) 

n = 1 n a 

The quantity of gas contained in the upper and lower tubes, after the elapse 
of the time t, is, respectively, 

r-2 ro 

Pl . dx ; Q" = q I a/2 a/2pj .dx,. . . (30) 

,vhere q is the sectional area of the tube. Hence show that 

- At/a* + I, - STrf/rf + . 

3^ 
from which the constant can be determined. If the time is sufficiently long, 



where D and S respectively denote the sum and difference of the quantity of 
gas contained in the two vessels. Loschmidt measured D, S, t, and a, and 
found that the agreement between observed and calculated results was very 
close. 

(8) The velocity of tJie solution of solids is a special case of diffusion. The 
layer of liquid in immediate contact with the solid is to be regarded as a 
saturated solution, the rate of solution thus depends upon the rate of the 
diffusion of the salt from the saturated solution to the adjoining layers of 
solvent. This problem can be attacked by the above method. For experi- 
mental work based upon the relation 

d ~ = qC(Q - x) ; or, \ log ^-^ = constant . . (32) 

(see Noyes and Whitney, Zeitschrift fiir physikalische CJwmie, 23, 6S9, 1897 : 
Bruner and Tolloczko, ibid., 35, 283, 1900) ; in this formula Q denotes the 
quantity of salt contained in a saturated solution, x the amount dissolved in 
the time t, q the area of the dissolving surface, C the velocity constant. 



385 



PART III. 

USEFUL EESULTS FKOM ALGEBEA AND 
TKIGNOMETEY. 

CHAPTEE IX. 

HOW TO SOLVE NUMERICAL EQUATIONS. 

155. Some General Properties of the Roots of Equations. 

THE solution of algebraic and transcendental equations is an im- 
portant branch of practical mathematics. The object of solving 
these equations is to find what value or values of the unknown 
will satisfy the equation, or will make one side of the equation 
equal to the other. Such values of the unknown are called roots 
or solutions of the equation. 

General methods for the solution of algebraic equations of the 
first, second and third degree are treated in regular algebraic text- 
books ; it is, therefore, unnecessary to give more than a brief 
resum& of their more salient features. 

R. N. Abel and Wantzel have brought forward demonstrations 
with the object of proving that general methods for the solution 
of equations of a higher degree than. the fourth are impossible. 
M'Ginnis has recently published a method which he claims can be 
employed for equations as high as the twelfth degree. 

Equations of higher degree than the fourth are comparatively 
rare in practical work.* Indeed we nearly always resort to the 
approximation methods for finding the roots of the numerical 
equations found in practical calculations. 

* Otherwise I should take advantage of the generosity of Professor M'Ginnis, and 
summarise his methods. They will, however, be found in The Universal Sulutwn, 1900 
(Swan, Sonnenschein & Co.). 

BB 



386 HIGHER MATHEMATICS. 155. 

The reader must distinguish between identical equations like 

(x + I) 2 = <r 2 + 2x + 1, 
which are true for all values of x, and conditional equations like 

z 2 + 2x + 1 = 0, 

which are only true when x has some particular value or values. In this case, 
for x = - 1. 

An equation like 

x 2 + 2x + 2 = 0, 

has no real roots because no real values of x will satisfy the equation. By 
solving as if the equation had real roots, the imaginary again forces itself on 
our attention. The imaginary roots of this equation are - 1 + \/ - 1, or - 1 + i. 

The general equation of the nth degree is 

x n + ax n " 1 + bx n ~ 2 + . . . + sx + E = 0. . (1) 
The term R is called the absolute term. If n = 2, the equation 
is a quadratic, x 2 + ax + E = ; if n = 3, the equation is said to 
be a cubic ; if n = 4, a biquadratic, etc. If x n has any coefficient, 
we can divide through by this quantity, and so reduce the equa- 
tion to the above form. When the coefficients a, b, . . ., instead 
of being literal, are real numbers, the equation is said to be 
numerical. 

The following synopsis of results proved in the regular text- 
books is convenient for reference : 

1. Every equation of the nth degree has n equal or unequal roots and no 
more (Gauss' law). E.g., 

x 5 + X* + x + 1 = 0, 
has five roots and no more. 

2. If an equation can be divided by x - a, without remainder, a is a root 
of the equation. More generally, if o, , 7, are the roots of an equation of the 
third degree, 

x s + a tf + b x + c = (x - a)(x - 0)(x - 7 ). . . (2) 

3. If the results obtained by substituting two numbers are of opposite 
signs, at least one root lies between the numbers substituted. 

4. An equation of an even degree, with its absolute term negative, has at 
least two real roots of opposite sign. 

5. An equation of an odd degree has at least one real root, the same in 
sign as the absolute term. 

6. Imaginary roots in an equation with real coefficients occur in pairs. 
E.g., if a + /8 \/ - 1 is one root of the equation, o - /3 V - 1 is another. 

7. The sum of the roots of an equation is equal to the coefficient - b of 
the second term ; the sum of the products of the roots taken two at a time is 
equal to + c ; the products of the roots taken three at a time is equal to - d, 
etc. ; the product of all the roots is equal to - (absolute term), if n is odd, and 
to + (absolute term), if n is even. 



< !..(.. HOW TO SOLVE NUMERICAL EQUATIONS. 387 

8. An equation, f(x), cannot have more positive roots than there are 
changes of sign in /( - x) (Descartes' rule of signs). E.g., in 

x + x*-x* + x+l = = f(x) ; - x + X s + x* - x + 1 = = /( - x), 
there are two changes of sign. Hence the equation has no more than three 
negative and two positive real roots. The remainder are imaginary roots. 

a. If the coefficients are all positive, the equation cannot have a positive 
root. Such is 

ar 6 + ar 5 + x + l=0. 

b. If the coefficients of the even powers of the unknown have the same 
sign, and the coefficients of the odd powers of the unknown have the opposite 
sign, the equation has no negative root. E.g., 

X l + >T 5 _ yA + X 3 _ X 2 + x _ I = 

c. If an equation has only even powers of x, with its coefficients all of the 
same sign, there is no real root. Thus, 

x 8 + or 4 + x 2 + 1 = 0. 

d. If the equation has only odd powers of x, with coefficients all of the 
same sign, there are no real roots other than x = 0. For instance, 

x 7 + x 5 + a* 3 + x = 0. 



156. The General Solution of Quadratic Equations. 

To recapitulate the results of the elementary textbooks : 

After suitable reduction, every quadratic may be written in the form : 

ax 2 + bx + c = ...... (1) 

If a and ft represent the roots of this equation, x must be equal to a or , 
where 

- b + \/6 2 - 4ac - b - \/b* - 4ac 



The sum and product of these roots are 

a + ft = - 6/a ; a/8 = c/. 

Hence if one of the roots is known, the other can be deduced directly. If 
a 1, the sum of the roots is equal to the coefficient of the second term with 
its sign changed, the product of the roots is equal to the absolute term. 
Equation (1), may be variously written 

a{x*- (a + 0)x + 0)8} = 0; 

a{x 2 - (Sinn of Roots)x + (Product of Roots)} = ; 

(.' - a) (x - 0) = ; a- 2 + -r + - = 0. 
a a 

From (2), we can deduce many important particulars respecting the nature of 
the roots of the quadratic. These are : 



388 



HIGHER MATHEMATICS. 



157. 



Relation between the Coefficients. 


The Nature of the Roots. 




/positive, 


real and unequal. 


(3) 


1 zero, . . . 


real and equal. 


(4) 


b 2 - 4ac is I negative, 


imaginary and unequal. 


(5) 


1 perfect square, 
Inot a perfect square, . 


rational and unequal. . 
irrational and unequal. 


(6) 
(7) 


a, b, c, have the same sign, 


negative. 


(8) 


a, 6, differ in sign from c, 


opposite sign. 


(9) 


a, c, differ in sign from 6, 


positive. 


(10) 


a=0, 


one root infinite. . 


(11) 


6 = 0, 


equal and opposite in sign. 


(12) 


c = 0, 


one root zero. 


(13) 


c = 0, b = 0, 


both roots zero. 


(14) 



In the table, the words " equal " and " unequal " refer to the numerical values 
of the roots. On account of the important role played by the expression 
6 2 - 4oc, in fixing the character of the roots, "6 2 - 4ac," is called the dis- 
criminant of the equation. 



157. Graphic Methods for the Approximate Solution of 
Numerical Equations. 

In practical work, it is generally most convenient to get ap- 
proximate values for the real roots of equations of higher degree 
than the second. Cardan's general method * for equations of the 
ohird degree, is generally so unwieldy as to be almost useless. 
Trignometrical methods are better. For the numerical equations 
pertaining to practical work, one of the most instructive methods 
for locating the real roots, is to trace the graph of the given 
function. Every point of intersection of the curve with the 
#-axis, represents a root of the equation. 

6 The location of the roots of the equation thus reduces itself to 
the determination of the points of intersection of the graph of the 
equation with the #-axis. 

EXAMPLES. (1) Find the root of the equation x + 2 = 0. At sight, of 
course, we know that the root is- - 2. But plot the curve y = x + 2, for 
values of y when - 3, - 2, - 1, 0, 1, 2, 3, are successively assigned to x. The 
curve (Fig. 122) cuts the or-axis when x = - 2. Hence, x = - 2, is a root of 
the equation. 

Another way is to proceed as in the next example. 

(2) Solve ar j + x - 2 = 0. Here X s = - x + 2. Put y = y? and y = - x + 2. 



* Found in the regular textbooks. 



jj 1.7T. HOW TO SOLVE NUMERICAL EQUATIONS. 



Plot the graph of each of these equations, using the table of cubes, page 518. 
The abscissa of the point of intersection of these two curves is one root of the 
given equation, x = OM (Fig. 123) is the root required. 




-y 

FiG. 122. 

(3) Locate the roots of x" 2 - 8x + 9 = 0. Proceed as before by assigning 
successive values to x. Roots occur between 6 and 7 and 1 and 2. 

(4) Show that or 3 - 6x 2 + Hx -6 = has roots in the neighbourhood of 
1, 2, 3. 

The method indicated in the second example, may be employed to find 
the roots of simultaneous equations, thus 

(5) Solve x 2 + ?/ = 1 ; x 2 - 4x = i/ 2 - By. 
Plot the two curves as shown in Fig. 124, 

hence x = + OM are the roots required. 

The graphic method can also be em- 
ployed for transcendental equations. 

(6) If x -f cos x = 0, we may locate the 
roots by finding the point of intersection of 
the two curves y - x and y = cos x. 

(7) If x + e x = 0, plot y = e* and y= - x. 
Table, page 518. 

(8) Show, by plotting, that an equation 
of an odd degree with real coefficients, has 
either one or an odd number of real roots. 

For large values of x, the graph must 
lie on the positive side of the cc-axis, and on 




FIG. 124. 



the opposite side for large negative values of x. Therefore the graph must 
cut the ar-axis at least once ; if twice, then it must cut the axis a third time, 
etc. 

(9) Show, by plotting, that an equation of an even degree with real co- 
efficients, has either 2, 4, ... or an even number of roots, or else no roots 
at all. 

(10) Prove, by plotting, (3), 155. 

(11) Plot x 2 - 2.r + 1 = 0. The curve touches but does not cut the .r-axis. 
This means that the point of contact of the curve with the x-axis, corresponds 
to two points infinitely close together. That is to say, that there are at least 
two equal roots. 

The graphic method may be applied to the most complicated equations. 



390 HIGHER MATHEMATICS. $ 158. 

(12) Find numbers, correct to three significant figures, which will satisfy 
the following equations : 

(i.) 9s 3 - 41x' 8 + 0-5 2 * - 92 = 0. (iii.) & - e~* + Q-lx - 10 = 0. 

(ii.) 2-42X 3 - 3-l5logeX - 20-5 = 0. (iv.) 2* 3 ' 1 - 3x - 16 = 0. 
(London S. and A. Depart., 1899 and 1900 Examinations.) Ansrs. (i.) 2-35 ; 
(ii.) 2-11; (iii.) 2-22; (iv.) 2-18. 

The accuracy of the graphic method depends on the scale of the diagram 
and the skill of the draughtsman. 

158. Newton's Method for the Approximate Solution of 
Numerical Equations. 

The above method indicates that the equation 

f(x) = y = x*-7x+l, . . (1) 

has a root lying somewhere between - 3 and - 4. We can keep 
on assigning intermediate values to x until we get as near to the 
exact value of the root as our patience will allow. Thus, if x= -3, 
y = + 1, if x = - 3'2, y = - 3'3. The desired root thus lies some- 
where between - 3 and - 3 '2. Assume that the actual value of 
the root is - 3'1. To get a close approximation to the root by 
plotting is a somewhat laborious operation. Newton's method, 
based on Taylor's theorem, allows the process to be shortened. 

Let a be the desired root, then 

/(a) = a 3 - 7a + 7. . . . (2) 

As a first approximation, assume that a = - 3-1 + h, is the required 
root. 

From (1), by differentiation, 

dy/dx = 3x 2 - 7 ; d*y/dj? = 6x ; d*y/dx* = 6. . (3) 
All succeeding derivatives are zero. 
By Taylor's theorem, 



Put v = - 3-1 and a = v + h. 

ft \ 4-( , i\ ft \ i>dv h' 2 d' 2 v h 

j()=j( + k)= /(V ) + h- rx + -.- + 

Neglecting the higher powers of h, in the first approximation, 



where /'(y) = dv/dx. The value of f(v) is found by substituting 
- 3'1, in (2), and the value of f(v), by substituting - 3 4 1, in the 
first of equations (3), thus, from (4), 

h =f(v)lf(v] = - 1-091/21-83 = - 0-04999. 
Hence the first approximation to the root is - 3-05. 



159. HOW TO SOLVK M'MKRICAL EQUATIONS. :;<H 

As a second approximation, assume that 

a = - 3-05 + h l = v l + h r 
As before, 

fcj = - /(oj/ffa) = - -022625/20-9081 = - -001082. 
The second approximation, therefore, is - 3-048918. We can, in 
this way, obtain third and higher degrees of approximation. Here 
is another example to try : 

x 3 - 2x - 5 = 0, 

has a root between 2 and 3. The first approximation is 2-0946, 
the second 2*09455148. Generally, the first approximation gives 
all that is required for practical work. 

EXAMPLES. (1) In the same way show that the first approximation to 
one of the roots of x* - x' 2 - 2x + 4 = 0, is a 4-2491 . . . and the second 
a = 4-2491405. . . . 

(2) If .i- 1 + 2x 2 + 3* - 50 = 0, x = 2-9022834. . . . 

(3) If x 2 + 4 sin x = 0, = - 1-933. . . . 

159. How to Separate Equal Roots from an Equation. 

This is a preliminary operation to the determination of the 
roots by a process, perhaps simpler than the above. 

If a, ft, y, . . . are the roots of an equation of the nth degree, 
x n + ax n ~ l + . . . + sx + E = 0, 

(*-.)(-0 . . . (x-r,) = 0. 

If two of the roots are equal, two factors, say x - a and x - ft, 
will be identical and the equation will be divisible by (x - a)- ; if 
there are three equal roots, the equation will be divisible by (x - a) 3 , 
etc. 

If there are n equal roots, the equation will contain a factor 
(x - a)", and the first derivative will contain a factor n(x - a)"" 1 , 
or x - a will occur n - 1 times. 

The highest common factor of the original equation and its 
first derivative must, therefore, contain x - a, repeated once less 
than in the original equation. If there is no common factor, there 
are no equal roots. 

EXAMPLES. (1) jr 5 - So: 2 - 8.1- + 48 = has a first derivative 3x 2 - lO.r - 8. 
The common factor is x - 4. This shows that the equation lias two roots 
equal to x - 4. 

(2) x 4 + Tx 3 - So,- 2 - 55.V + 50 = has two roots each equal to x + 5. 



392 HIGHER MATHEMATICS. 160. 

160. Sturm's Method of Locating the Real and Unequal 
Roots of a Numerical Equation. 

Newton's method of approximation does not give satisfactory 
results when the two roots have nearly equal values. For instance, 
the curve 

y = x 3 - Ix + 7 

has two nearly equal roots between 1 and 2, which do not appear 
if we draw the graph for the corresponding values of x and y, viz.: 
x = 0, 1, 2, 3, . . . ; 
y = 7, 1, 1, 13, ... 

The problem of separating the real roots of a numerical equa- 
tion is, however, completely solved by what is known as Sturm's 
theorem. It is clear that if x assumes every possible value in 
succession from + oo to - oo, every change of sign will indicate 
the proximity of a real root. The total number of roots is known 
from the degree of the equation, therefore the number of imaginary 
roots can be determined by difference. 

Number of real roots + Number of imaginary roots = Total number of roots. 

Sturm's theorem enables these changes of sign to be readily 
detected. The process is as follows : 

First remove the real equal roots, as indicated in the preceding 
section, let 

y = x* - Ix + 7, . . (1) 

remain. Find the first differential coefficient, 

dy/dx = 3x 2 - 7 (2) 

Divide the primitive (1) by the first derivative (2), thus, 

(x 3 -Ix + 7)/(Sx 2 - 7). 
Change the sign of the remainder and divide by 7, the result 

R = Zx - 3, . . . . (3) 

is now to be divided into (2). Change the sign of the remainder 
and we obtain, 

B-l (4) 

The right-hand sides of equations (1), (2), (3), (4), 

y? - Ix + 7; 3x 2 - 7; 2x - 3; 1, 
are known as Sturm's functions. 

Substitute - oo for x in (1), the sign is negative ; 
(2), ,, positive; 

,, ,, (3), ,, negative; 

(4), positive. 



h,n. HOW TO SOLVE NUMERICAL EQUATION-. 393 

Note that the last result is independent of x. The changes of 
sign may, therefore, be written 

In the same way, 



Value of x. 


Corresponding Signs 
of Sturm's Functions. 


Number of Changes 
of Sign. 


- CD 


- + - + 


3 


_ ^ 


- + - + 


3 


- 3 


+ + - + 


2 


- 2 


+ + - + 


2 


- 1 


+ - - + 


2 


+ 


+ -- - + 


2 


+ 1 


+ - - + 


2 


+ 2 


f + + + 





+ oo 


+ + + + 






There is, therefore, no change of sign caused by the substitution 
of any value of x less than - 4, or greater than + 2 ; on passing 
from - 4 to - 3, there is one change of sign ; on passing from 
1 to 2, there are two changes of sign. The equation has, there- 
fore, one real root between - 4 and - 3, and two, between 1 and 2. 

It now remains to determine a sufficient number of digits, to 
distinguish between the two roots lying between 1 and 2. First 
reduce the value of x in the given equation by 1. This is done by 
substituting u + 1 in place of x, and then finding Sturm's functions 
for the resulting equation. These are, 

U 3 + 3^2 _ u + i . 3^2 + 6w - 4 ; 2 - 1 ; 1. 
As above, noting that if x + 1, u = + O'l, etc., 



Value of x. 


Corresponding Signs 
of Sturm's Functions. 


Number of Changes 
of Sign. 


1 


+ - - + 


2 


2 


+ - - + 


2 


3 


+ - - + 


2^ 


4 


_ I 


1 


5 


- - + + 


1 


6 


- + + + 


li 


7 


+ + + + 






The second digits of the roots between 1 and 2 are, therefore, 
3 and 6, and three real roots of the given equation are approxi- 
mately - 3, 1-3, 1-6. 



394 HIGHER MATHEMATICS. .< lio 

EXAMPLES. Locate the roots in the following equations : 

(1) x* - 3a- 2 - 4.r + 13. Ansr. Between - 3 and - 2 ; 2 and 2 -5 ; 2-5 and 3. 

(2) or* - 4o.- 2 - 6x + 8. Ansr. Between and 1 ; 5 and 6 ; - 1 and - 2. 

(3) x 4 + or - ic 2 - 2x + 4. We have five Sturm's functions for this equa- 
tion. Call the original equation (1), the first derivative, Ix 3 + 3x* - 2z - 2, (2) ; 
divide (1) by (2) and a- 2 + 2x - 6 (3) remains ; divide (2) by (3) and - x + 1 (4) 
remains ; divide (3) by (4) and change the sign of the result for + 1 (5). Now 
let x = + oo and - oo, we get 

+ + -\ h (2 variations of sign) : H f- + + (2 variations). 

This means that there are no real roots. All the roots are imaginary. 

(4) The equation, 3? - 3rx 2 + 4^ = 0, is obtained in problems referring 
to the depth to which a floating sphere of radius r and density p sinks in 
water. Solve this equation for the case of a wooden ball of unit radius and 
specific gravity 0'65. Hence, x 3 - Sx + 2'6 = 0. The three roots, by Sturm's 
theorem, are a negative root, a positive root between 1 and 2, and one over 2. 
The depth of the sphere in the water cannot be greater than its diameter 2. 
The negative root has no physical meaning. These two roots must, therefore, 
be excluded from the solution. The other root, by Newton's method of 
approximation, is x = 1-204. . . . 

In this last example we have rejected two roots because they were incon- 
sistent with the physical conditions of the problem under consideration. This 
is a very common thing to do. Not all the solutions to which an equation 
may lead are solutions of the problem. Of course every solution has some 
meaning, but this may be quite outside the requirements of the problem. 
Imaginary roots may be obtained, when the problem requires real numbers, 
the roots may be negative or fractional, when the problem requires positive 
or whole numbers. Sometimes, indeed, none of the solutions will satisfy the 
conditions imposed by the problem, in this case the problem is indeterminate. 

To illustrate : 

1. A is 40 years, B 20 years old. In how many years will A be three 
times as old as B ? Let x denote the required number of years. 

.-. 40 + x = 3(20 + x) ; or x = - 10. 

But the problem requires a positive number. The answer, therefore, is that 
A will never be three times as old as B. (The negative sign means that A 
was three times as old as B, 10 years ago.) 

2. A number x is squared ; subtract 7 ; extract the square root of the 
result ; add twice the number, 5 remains. What was the number x? 

.-. 2z + x /(z 2 - 7) = 5. 

Solve in the usual way, namely, square 5 - 2x = \'.r 2 - 7 ; rearrange terms 
and use (2), 156. Hence x = 4 or |. 

The ultimate test of every solution is tliat it shall satisfy the equation when' 
substituted in place of the variable. If not it is no solution. On trial both 
solutions, x = 4 and x = 2|, fail to satisfy the test. These extraneous solutions 
have been introduced during rationalisation (by squaring). 



< n;i. HOW TO SOLVE NUMERICAL KnTATloNs. :;;, 

161. Horner's Method for Approximating to the Real 
Roots of Numerical Equations. 

When the first significant digit or digits of a root have been 
obtained, by, say, Sturm's theorem, so that one root may be 
distinguished from all the other roots nearly equal to it, Horner's 
method is one of the simplest and best ways of carrying the 
approximation as far as may be necessary. So far as practical 
requirements are concerned, Horner's process is perfection. The 
arithmetical methods for the extraction of square and cube roots 
are special cases of Horner's method, because to extract \9, or 
x/9, is equivalent to finding the roots of the equation # 2 - 9 = 0, 
or a 3 - 9 = 0.* 

In outline, the method is as follows : Find by means of Sturm's 
theorem, or otherwise, the integral part of a root, and transform 
the equation into another whose roots are less than those of the 
original equation by the number so found. Suppose we start 
with the equation 

a; 8 - Ix + 7 - 0, . . . (1) 

which has one real root whose first significant figures we have 
found to be 1'3. Transform the equation into another whose 
roots are less by 1-3 than the roots of (1). This is done by 
substituting u + 1*3 for x. In this way we obtain, 

u + 3-95%' 2 - 1-93M + '097 - 0. . . (2) 
The first significant figure of the root of this equation is '05. Lower 
the roots of (2) by the substitution of v + '05 for u in (2). Thus, 
v 3 + 4-05i7 2 - 1-53250 + -010375 - 0. . . (3) 
The next significant figure of the root, deduced from (3), is '006. 
We could have continued in this way until the root had been 
obtained of any desired degree of accuracy. 

* Chrystal, Textbook of Algebra ( A. & C. Black, London, 1898, Part I., page 346), 
s;iys : " Considering the remarkable elegance, generality, and simplicity of the method, it 
is not a little surprising that it has not taken a more prominent place in current mathe- 
matical textbooks. Although it has been well expounded by several English writers, 
... it has scarcely as yet found a place in English curricula. Out of five standard 
Continental textbooks where one would have expected to find it we found it mentioned 
in only one, and there it was expounded in a way which showed little insight into its 
true character. This probably arises from the mistaken notion that there is in the 
method some algebraic profundity. As a matter of fact, its spirit is purely arith- 
metical ; and its beauty, which can only be appreciated after one has used it in 
particular cases, is of that indescribably simple kind which distinguishes 'the use of 
position in the decimal notation and the arrangement of the simple rules of arithmetic. 
It K in short, one of those things whose invention was the creation of a commonplace." 



396 HIGHER MATHEMATICS. $ 161. 

Practically, the work is not so tedious as just outlined. Let a, b, c, 
be the coefficients of the given equation (1), E the absolute term. 

1. Multiply a by the first significant digits of the root and add 
the product to b. Write the result under b. 

2. Multiply this sum by the first figures of the root, add the 
product to c. Write the result under c. 

3. Multiply this sum by the first figure of the root, add the 
product to E, and call the result the first dividend. 

4. Again multiply a by the root, add the product to the last 
number under b. 

5. Multiply this sum by the root and add the product to the 
last number under c, call the result the first trial divisor. 

6. Multiply a by the root once more, and the product to the 
last number under b. 

7. Divide the first dividend by the first trial divisor, and the 
first significant figure in the quotient will be the second significant 
of the root. Thus starting from the old equation (1), whose root 
we know to be about 1, 

a b c R (Root 

1 +0 -7 +7 (1-3 

1 1 - 6 

1-6 1 First dividend. 

1 2 



2 - 4 First trial divisor. 

1 

3 

8. Proceed exactly as before for the second trial divisor, using 
the second digit of the root, viz., -3. 

9. Proceed as before for the second dividend. We finally ob- 
tain the result shown in the next scheme. Note that the black 
figures in the preceding scheme are the coefficients of the second 
of the equations reduced on the supposition that x = 1-3 is a root 
of the equation. 

a' b' c' R' (Root. 

13 - 4 1 (1-35 

0-3 0-99 - 0-903 

3-3 - 3-01 0-097 Second dividend. 

0-3 1-08 

3-6 - 1-93 Second trial divisor. 

0-3 

3-9 



HH. HOW TO SOLVE NUMERICAL EQUATION- :i'.)7 



b" 

3-9 
0-05 


c" 
- 1-93 
0-1975 


R" 
0-097 
- 0-086625 


(Root 
(1-356 


3-95 
0-05 


- 1-7325 
0-2000 


0-010375 Third dividend. 
Third trial divisor. 


4-00 
0-05 


- 1-5325 






4-05 

Having found about five or seven decimal places of the root in 
this way, several more may be added by dividing, say the fifth 
trial dividend by the fifth trial divisor. Thus, we pass from 
1-356895, to 1-356895867 ... a degree of accuracy more than 
sufficient for any practical purpose. 

Knowing one root, we can divide out the factor x - 1-3569 from 
equation (1), and solve the remainder like an ordinary quadratic. 

If any root is finite, the dividend becomes zero, as in one of 
the following examples. If the trial divisor gives a result too large 
to be subtracted from the preceding dividend, try a smaller digit. 

To get the other root whose significant digits are 1-6, proceed 
as above, using 6 instead of 3 as the quotient from the first dividend 
and trial divisor. Thus we get 1-692 . . .* 

It is usual to write down the successive steps as indicated in 
the following example. 

EXAMPLES. (1) Find the root between 6 and 7 in 
4X 3 - IBx 2 - Six = 275. 



4 - 13 
24 


- 31 

66 


- 275 (6-25 
210 


11 
24 


35 
210 


- 65 

51-392 


35 

24 


245 

11-96 


- 13-608 

13-608 


59 

0-8 


256-96 
12-12 





59-8 
0-8 


269-08 

3-08 




60-6 
0-8 


. 272-16 




61-4 







* Several ingenious short cuts have been devised for lessening the labour in the 
application of Horner's method, but nothing much is gained, when the method has 
only to be used occasionally, beyond increasing the probability of error. 



398 HIGHER MATHEMATICS. 162. 

The steps mark the end of each transformation. The digits in block 
letters are the coefficients of the successive equations. 

(2) There is a positive root between 4 and 5 in X s + x* + x - 100. Ansr. 
4-2644 . . . 

(3) Find the positive and negative roots in 

x* + So- 2 + 16o- = 440. 
Ansr. + 3-976 . . ., - 4-3504 . . . 

When finding negative roots, proceed as before, but first transform the equa- 
tion into one with an opposite sign by changing the sign of the absolute term. 

(4) Show that the root between - 3 and - 4, in equation (1), is 3-0489173396 
Work from a = l, 6 = - 0, c = - 7, R = - 7. 



162. van der Waals' Equation of State. 

The relations between the roots of equations, discussed in this 
chapter, are interesting in many ways ; for the sake of illustration, 
let us take van der Waals' equation of state for a gas at a distance 
from its point of liquefaction, 

-*) = **. . . . (1) 



or, expanded, v* - (b + V + -t? - = 0. (2) 

p) p p 

This equation of the third degree in v, must have three roots, 
<x, ft, y, equal or unequal, real or imaginary. In any case, 

(V - a)(v - fi(v - y) = Q. . (3) 

Imaginary roots have no physical meaning ; we may therefore 
confine our attention to the real roots. Of these, we have seen 
that there must be one, and there may be three. This means 
that there may be one or three (different) volumes, corresponding 
to every value of the pressure j9 and temperature 6. There are three 



interesting cases : 



Case i. There is only one real root present. This implies that 
there is one definite volume (v) corresponding to every assigned 
value of pressure (p) and temperature (0). This is realised in the 
pv-GurvGj for all gases under certain physical conditions ; for in- 
stance, the graph for carbon dioxide at 48-1 (Fig. 125), has only 
one value of p corresponding to each value of v. 

The collection of curves shown in Fig. 125, were obtained by 
plotting values, of p and v corresponding to different values of 6, 
a and b. In the diagram, the degrees are on the centigrade scale. 
In reality, 

= (273 + degrees centigrade). 



jj HW. HOW TO SOLVE NUMERICAL EQUATIONS. 




Case ii. T/iere arc three real unequal roots present. For tem- 
peratures below 32*5, say 13 '1, we get the wavy curve ABCD 
(Fig. 125). This means that at 13 '1, and at a pressure of Op , 
carbon dioxide ought to have 
three different volumes corre- 
sponding with the abscissae 
Oy, Oft, Oa. Only two of 
these three volumes have yet 
been observed, namely for 
gaseous C0 2 at a and for liquid 
CO., at y, the third, correspond- 
ing to the point (3, is unknown. 
The curve AyfiaD, has been 
realised experimentally by 
Andrews. 

When the volume of a 
mass of carbon dioxide gas is 
gradually diminished, the cor- 
responding changes of pressure 
and volume are represented 
graphically by the curve Da. 
At the point a, the gas begins 
to condense ; continuing the lessening of the volume, the pressure 
remains constant, until the point y is reached. Here, all the 
carbon dioxide will have assumed the liquid state. The straight 
line ya thus represents the constant pressure exerted by the vapour 
of carbon dioxide in contact with its liquid. 

The steep curve Ay indicates that there is only a slight change 
in the volume of the liquid for great increments of pressure. See 
13. 

The abscissa of the point a represents the volume of a given 
mass of gaseous carbon dioxide, the abscissa of the point y 
represents the volume occupied by the same mass of liquid carbon 
dioxide at the same pressure. 

Under special conditions, parts of the sinuous curve yBftCa 
have been realised experimentally, namely, yx and ay. These 
latter represent unstable conditions of supersaturation. The por- 
tion yx shows that a liquid may exist at a pressure less than that 
of its own vapour, and ay shows that a vapour may exist at a pres- 
sure higher than that of its " vapour pressure " of its own liquid. 



FIG. 125. Isothermals of Carbon 
Dioxide. 



400 HIGHER MATHEMATICS. 102. 

Case iii. There are three real equal roots present. At and above 
the point where a = ft = y, there can only be one value of v for any 
assigned value of p. This point is no other than the well-known 
critical point of a gas. Write p c , v c , C , for the critical pressure, 
volume and temperature of a gas. From (3), 

(v - a)* = 0, . . . (4) 

and at the critical point a = v = v c , therefore, 
-R0A a ab 

+ P./ + ?>c V ~ Yc = 

This equation is an identity, therefore (footnote, page 172), 

3v c p c = bp c + R6 C ; 3v 2 c p c = a ; v* c p c = ab, . (6) 
are obtained by equating the coefficients of like powers of the 
unknown v. 

From the last two of equations (6), 

v e = 3b. ... . (7) 

From (7) and the second of equations (6), 

P--W& < 8 > 

From (7), (8), and the first of equations (6), 



From these results, (7), (8), (9), van der Waals has calculated the 
values of the constants a and b for different gases. 

Let TT = p/p c , <fr = v/v c , 0' = 0/0* From (1), (7), (8) and (9), 

(TT + 3/< 2 ) (30 - 1) = SO', . . . (10) 

which appears to be van der Waals' equation freed from arbitrary 
constants. This result has led van der Waals to the belief that 
all substances can exist in states or conditions where the corre- 
sponding pressures, volumes and temperatures are equivalent. 
These he calls corresponding states ( tf Uebereinstimmende Zu- 
stande "). The deduction has only been verified in the case of 
ether, sulphur dioxide and some of the benzene halides. 

It is an interesting exercise to apply the methods of Chapter III. to the 
"singular points" of the curves shown in Fig. 125. For convenience, put 
RB = constant, say c. Solve (1) for p, 

p = c/(v - b) - a/y 2 (11) 

Differentiate twice, 

dp _ c 2a d?p 2c 6a 

dv~~ (v - 6) 2 + v 3 ' dv* ~ (v - ft) 3 ~ v 4 * 

Now read over 55 to 60. It is not difficult to see that if the tempera- 
ture 6 is high enough, dpjdv is always negative, that is to say, the curve, or 






162. HOW TO SOLVE NUMERICAL EQUATIONS. 401 

rather its tangent, will slope from left to right like the hyperbola 48-1 (Fig. 
125). If v is small enough, so that v - b approaches zero, the curve will have 
a negative slope like the left sides of the curves 32-5 to 19-1 (Fig. 125), because 
dpjdv will still remain negative. 

When dpjdv becomes zero, we may have either 

(a) a point of inflection shown in curve 32*5 (Fig. 125) ; or 

(6) maximum and minimum values indicated by the dotted lines in 

the 19-1 curve. 

When 6 is small enough we may have, for certain values of v, a positive 
value for dp/dv. This can only correspond to the slope of the dotted portion 
BC of the curve 19-1 (Fig. 125). 

Now'rearrange the first of equations (12), so that 



..... (13) 

When v = 6, (v - 6) 2 /u 3 = ; when v = 36, this expression reaches a 
maximum and gradually diminishes to zero as v approaches oo. If c is 
greater than 8a/276, c, or what is the same thing, R0, is greater than the 
maximum of 2a(v - 6) 2 /tr 3 , therefore, as v increases p decreases. When c is 
less than 8a/276, p decreases for small and large values of v ; p only increases 
in the neighbourhood of v = 36. The expression has thus a maximum or a 
minimum value for any value of v which makes 2a(v - b) 2 /v 3 = R0. 

Equating the second differential coefficient, in (12), to zero, we get 



_ = a ^ 

c c c c 

The roots of this biquadratic in v, correspond with the points of inflection or 
transition points of the curve. Of these, there may be four, two, or none. 

Now try and plot van der Waals' equation for any gas from the published 
values of a, 6, R. For example, for ethylene a = 0*00786, 6 = 0-0024, R = 0-0037 ; 
for carbon dioxide 

p + ' 00874 \(t; - 0-0023) = 0-00369(273 + ,), . . (15) 



where 6 l denotes degrees of temperature on the centigrade scale. Hint. First 
fix the value for 1} say, 0C., and calculate a set of corresponding values of p 
and v, thus, 

v = 0-1, 0-05, 0-025, 0-01, 0-0075, 0-005, 0-004, 0-003, . . . ; 
p = 9-4, 19-7, 30-3, 43-3, 37-9, 23-2, 45-8, 466-8, . . . 
Make the successive increments in v small when in the neighbourhood of a 
singular point. Plot these numbers on squared paper. Note the points of 
inflection. Now do the same thing with e l = 30 C., and : = 50 C. In this 
way you will get a better insight into the " inwardness " of van der Waals' 
equation than if pages of descriptive matter were appended. Notice that the 
/w 2 term has no appreciable influence on the value of p when v becomes very 
great, and also that the difference between v and v - b is negligibly small, as 
r becomes very large. What does this signify ? When the gas is rarefied, it 
will follow Boyle's law pv = constant. What would be the state of the gas 
when v = 0-0023 ? 

See Hilton, Phil. Mag. [6], 1, 579 ; ib. t 2, 108, 1901. 

CC 



402 



CHAPTEE X. 
DETERMINANTS. 

THIS chapter is for the purpose of explaining and illustrating a 
system of notation which is in common use in the different branches 
of pure and applied mathematics. 

163. Simultaneous Equations. 

(i.) Homogeneous simultaneous equations in two unknowns. The 
homogeneous equations, 

a^x + % = ; a 2 x + b 2 y = 0, . (1) 

represent two straight lines passing through the origin. In this 
case ( 28), x = and y = 0, a deduction verified by solving for 
x and y. Multiply the first of equations (1) by b 2 , and the second 
by b r Subtract. Or, multiply the second of equations (1) by a lt 
and the first by a. 2 . Subtract. In each case, we obtain, 

x(aJ 2 - o^) = ; y(a 2 b l - aJJ = 0. . . (2) 
Hence, x = ; and y = ; 

or, a-J} 2 - ajb^^ = ; and ajb-^ - a l b 2 = 0. . . (3) 

The relations in equations (3) may be written, 

; and I a 2 , b 2 1 = 0, . (4) 



a 2 , b 2 a lt 

where the left-hand side of each expression is called a determinant. 
This is nothing more than another way of writing down the differ- 
ence of the diagonal products.* 

The products a^b^, a 2 6 x , are called the elements of the determinant ; 
!, &J, a 2 , 6 2 , are the constituents of the determinant. Commas may or may 
not be inserted between the constituents of the horizontal rows. When only 
two elements are involved, the determinant is said to be of the second order. 

* In literal equations, the letters should always be taken in cyclic order so that 
b follows a, c follows b, a follows c. In the same way 2 follows 1, 3 follows 2, and 
1 follows 3. 



< !;:;. DETERMINANTS. 403 

From the above equations, it follows that only when the de- 
terminant of the coefficients of two homogeneous equations in x and 
y is equal to zero can x and y possess values differing from zero. 

(ii.) Linear and homogeneous equations in three unknowns. 
Solving the linear equations 

a^x + b$ + c l = ; a, 2 x + b. 2 y + c. 2 = 0, . . (5) 
for x and ?/, we get 



= 

a A - *A" 

If a^ - b^ = 0, x and y become infinite. In this case, the two 
lines represented by equations (5) are either parallel or coincident. 

When x = bl l o ^ = oo ; y = ^ifi^Ml = 00f the lines in- 

tersect at an infinite distance away. Reduce equations (5) to the 
tangent form ( 30), 

a, c l a. 2 c. 7 

ni _ _ int _ i ni _ _ &rp _ ^ * 

~ b? ~ b, ' y - - b* - b. 2 ' 

but since a^ - b^ = 0, a l fb l = a. 2 /b> 2 = the tangent of the angle 
of inclination of the lines ; in other words, two lines having the 
same slope towards the #-axis are parallel to each other.* 

When the two lines cross each other, the values of x and y in 
(6) satisfy equations (5). Make the substitution required. 
a i( b i c 2 ~ b -2 c i) + b }( c i a -2 ~ c -2 a i) + G i( a A ~ a A) = 0, 

a -2( b l C >2 - b '2 C l) + b -2( C i a 2 ~ C 2 a i) + C 2( a A ~ a A) = > 

or, writing x = X/Z and y = Y/Z, ... (8) 

we get a pair of homogeneous equations in X, Y, Z, namely, 

a^X + b^Y + c^Z = ; a. 2 X + b. 2 Y +c. 2 Z = 0. . (9) 
Equate coefficients of like powers of the variables in these identical 
equations. 

.-. a l :b l :c l = a. 2 :b. 2 :c 2 , 
or, from (8) and (6), 

X: Y: Z = ^c. 2 - b. 2 c l : c^a* - c 2 a x : a^ - a. 2 b v 

= I b i c i I : I c i a i \ \ a i b i 
\b 2 c 2 \ |c 2 a 2 \ \a 2 b 2 

The three determinants on the right, are symbolised by 



* Thus the definition, " parallel lines meet at infinity," means that as the point 
of intersection of two lines goes further and further away, the lines become more and 
more nearly parallel. 



404 HIGHER MATHEMATICS. S 163. 

where the number of columns is greater than the number of rows.* 
The determinant (11), is called a matrix. It is evaluated, by 
taking the difference of the diagonal products of any two columns. 
The results obtained in (10) are employed in solving linear 
equations. 

EXAMPLES. (1) Solve 4* + 5y = 7 ; 3x - Hty = 19. 

X: Y:Z = \ 5, - 71:1- 7, 41:14, 51; 

1-10, - 19 I 1-19, 3 | I 3, - 10 | 
= - 165 : 55 : - 55 ; or x = + 3 and y = - 1. 

(2) Solve 20x - I9y = 23 ; 19x - 20y = 16. Ansr. x = 4, y = 3. 

(3) Solve the observation equations : 

5x - -2y = -4 ; '14i + -Sy = 1-18. Ansr. x = 2, y = 3. 

(4) Solve \x - \y = 6 ; $x - \y = - 1. Ansr. x = 24, y = 18. 

The condition that three straight lines represented by the 
equations 

a^x + \y + Cj = ; a 2 x + b 2 y + c. 2 = ; a.jc + b s y + c 3 = 0, (12) 
may meet in a point, is that the roots of any two of the three 
lines may satisfy the third ( 32). In this case we get a set of 
simultaneous equations in X, Y, Z. 

a 1 X+b 1 Y + c l Z = a 2 X + b. 2 Y+c 2 Z = a 3 X+bsY + c 2 Z = Q, (13) 
by writing x = X/Z and y = YjZ in equations (12). 

From the last pair, 

2 , b. 2 I. . (14) 



But these values of x and y, also satisfy the first of equations (3), 
hence, by substitution, 

C 2 1 + b i 

C 3J 

which is more conveniently written 

a, b, c, I = 0, . . . . (16) 

a 2 b 2 c 2 1 

a 3 b s c 3 I 
a determinant of the third order. 

It follows directly from equations (13), (14), (16), only when 
the determinant of the coefficients of three homogeneous equations 
in x, y, z, is equal to zero, can x, y, z, possess values differing from 
zero. 



* It is customary to call the vertical columns, simply " columns" ; the horizontal 
rows, " rows". 



^ H>4. DETERMINANTS. M 

From (12), (13), (15), (16), we conclude that three equations 
are consistent with each other, only when the. determinant of the 
coefficients and absolute term of three linear equations in x, y, z, 

<[ual to zero. 

This determinant is called the eliminant of the equations. 
Instead of taking the last two of equations (12), we might have 
substituted the values of x and y derived from any two of these 
equations in the third. Thus, in addition to (14), we may have 
X'.Y-.Z = 6 3 c 3 |:|c 3 a,\Aa, 6 3 

b i c i\ \ c i a i\ \ a i b i 
b i c i\\ c i a i\'-\ a i b i 

b -2 C 2\ \ C 2 a 2 I I a 2 b 2 

Each of these sets may be obtained from (16), by deleting certain 
rows and columns, for instance, I 6 3 c 3 1 is obtained by omitting 

K M 

the row and column containing a. 2 , and so on. Each determinant 
in (14) and (17), is called a subdeterminant, or minor of (16). 

164. The Expansion of Determinants. 

It follows from (15) and (16), that 



(17) 



a. 2 b. 2 c 2 \ 

3 b S C 3\ 

A determinant is expanded, by taking the product of one letter 
in each horizontal row with one letter from each of the other rows. 
The first element, called the leading element, is the product of 
the diagonal constituents from the top left-hand corner, i.e., a^c.^ ; 
its sign is taken as positive. The signs of the other five terms,* 
are obtained by arranging alphabetically, and observing whether 
they can be obtained from the leading element by an odd or an 
even number of changes in the subscripts ; if the former, the 
element is negative, if the latter, positive. For example, 2 ^ 1 c 3 , 
is obtained by one interchange of the subscripts 2 and 1 in the 
leading element ; 2 ^i c 3 ^ s > therefore, a negative element ; a.^b^Cj^ 
requires two such transformations, 2 and 1, and 2 and 3, hence 
its sign is positive. 

* The number of constituents in a determinant of the second order is 2 x 1, or 2 ! ; 
of the third order 3 x 2 x 1, or 3 !, of the fourth order, 4 !, etc. 



406 



HIGHER MATHEMATICS. 

1 + 12-8-4-6 = 4. 



S 165. 



EXAMPLES. (1) Show 12 2 21 = 2 
J3 1 1 I 
I ! 2 1 I 

(2) Show 10 b c I = 2abc. 

\b 
| (l I 

(3) Expand j a lt b^ Cj, d^ I into twenty-four terms, twelve negative, twelve 



positive. 



165. The Solution of Simultaneous Equations. 

Continuing the discussion in 162, let the equations 
a^x + b$ + c-^z = d^ ; a 2 x + b 2 y + c 2 z = d. 2 ; a z x + b B y + c%z = d%, (18) 
be multiplied by suitable quantities, so that y and z may be elimi- 
nated. Thus multiply the first equation by A^ the second by A 2 , 
the third by A z , where A^ A. 2 , A s , are so chosen that 

b^ + b. 2 A 2 + b. 3 A 3 = ; c l A 1 + c 2 A 2 + c 3 A 3 = 0. (19) 
Hence, by substitution, 

x(a 1 A l + a 2 A 2 + a s A 3 ) = d 1 A l + d 2 A 2 + d z A 3 . . (20) 
Equations (19) being homogeneous in A lt A 2 , A 3 , we get, from (10), 



A 1 : A 2 : A. 3 = I b. 2 b 



, b, 



Substituting these values of A lt A 2 , A 3 , in equations (20), we get, 
as in equations (14), (15), (16), 



x I a i b i c i\ = 



(21) 





u/2 t/2 ^2 

a 3 6 3 c 3 




2 2 2 

3 o 3 c 3 




In the same way, on multiplying by B v B 2 , B s , and by C lt C 2 , C 3 . 


y 


a i b i c i 


= 


j rfj C x 


; s 


a i b l C l 


= 


! 6 X ^j 


.(22) 




a 2 b 2 C 2 




2 d 2 c 2 




a. 2 b 2 c. 2 




a 2 b 2 d 2 






3 3 3 




tto ^o C, 
o o o 




a z b 3 C 3 




333 





EXAMPLES. Solve the following sets of equations : 

(1) 5x + 3y + 3z = 48 ; 2x + 6y - 3z = 18 ; 8x - By + 2z = 21. From (21) 



48 3 31+ 
18 6-3| 
21 -3 2| 



5 3 3 
2 6-3 
8-3 2 



similarly y = 5', z = 6. 

(2) x - m/ + a 2 2 = a 3 ; x - by + b 2 z = 6 3 ; x - cy + c*z = c 3 . Ansr. x = abc ; 
y = ab + be + ca ; z = a + 6 + c. 

(3) Solve 2a: - 3# + 4,s = 1 ; 3x + 2y - 4-z = ^ ; 4* - 3# + 2s = . Ansr. 
= i y = i, = } 



KM. 



DETERMINANT-. 



407 



(4) Solve the observation equations : 
-Sx + -2y + '5z = 8-2 ; -2x + "3y + - 
Ansr. x = 2, y = 3, z = 4. 



= 3'7. 



166. Elimination. 

It is required to eliminate the unknown from the two equations, 
a#? + a^c 2 + a^x + a = ; b. 2 x' 2 + b^x + b (} = 0. (23) 
Multiply the first equation by x, the second by x and by x 1 
successively. We thus get the five equations, 

+ a. 3 x? + a 2 x 2 + a^ + a = 0,^1 
a^ + a.jX 2 + a-^x 2 + a^x -f = 0, 



+ + b 2 x 2 + b^ + b (> = 0, J-. 
+ b. 2 x? + b.x 2 



b.x 2 + b x + = 0, 
b#? + ' + = 0,J 



(24) 



As on page 405, if these five equations are consistent, the eliminant 
of the four unknowns, is 

(25) 
" 
6 k b 



a, 







b n 



EXAMPLES. (1) Show that the following equations are consistent with 
one another, 

x + y - z = ; x - y + z = "2; y + z - x = 4 ; x + y + z = 6, 
11-10 
1-112 
-1114 
1116 
(2) Eliminate x and y from the equations 

Divide the first by y", the second by y-. Multiply the first by x/y, the second 
by xfy and x 2 jy~. The eliminant of the resulting five equations , is 

2-5 0-9 

2-5 0-9 

3-7-6 

3-7-6 

3 _7 _6 



167. Fundamental Properties of Determinants. 

1. The value of a determinant is not altered by charujiny the 
columns into rows, or the rows into columns. 



408 



HIGHER MATHEMATICS. 



167. 



It follows directly, by simple expansion, that 
and 



K &i| = |i 4 1; 



(26) 



2 C 2 I 0, 2 3 

, 6 3 cj |c : c 2 c 3 
It follows as a corollary, that whatever law is true for the rows of a 
determinant, is also true for the columns and conversely. 

2. The sign, not the numerical value, of a determinant is altered 
by interchanging any two rows, or any two columns. 

By direct calculation, 

b i c i I = ~ I 6 i i c x I. . (27) 

3. If two rows or two columns of a determinant are identical, 
the determinant is equal to zero. 

If two identical rows or columns are interchanged the sign, not the value 
of the determinant, is altered. This is only possible if the determinant is 
equal to zero. The same thing can be proved by the expansion of, say, 



a 3 a s c 5 

4. When the constituents of two roivs or two columns differ by a 
constant factor, the determinant is equal to zero. 



Thus by expansion show that 



415 

826 

12 3 7 



= 4 



1 1 51 = 4x0 = 0. 
2261 

3371 



(28) 



5. If a determinant has a row or column of cyphers it is equal 
to zero. 



This is illustrated by expansion, 

10 6 X Cl 
6 2 c 



(29) 



6. In order to multiply a determinant by any factor, multiply 
each constituent in one row or in one column by this factor. 



This is illustrated by the expansion of the following : 



ma s b, c s | 



(30) 



7. In order to divide a determinant by any factor, divide each 
constituent in one row or in one column by that factor. 



DETERMINANTS. 



409 



This follows directly from the preceding proposition. It is conveniently 
used in the reduction of determinants to simpler forms. Thus, 

6 9 81 = 9.6.211 1 41 = 9.6.2.211 1 21. . . (31) 
12 18 4 I I 2 2 2 I I 1 1 1 I 

24 27 2 I I 4 3 1 I | 4 3 1 1 

8. If the sign of every constituent in a row or column is changed, 
the sign of the determinant is changed. 

9. One row or column of any determinant can be reduced to 
unity (Dos tor's theorem). 



This will need no more explanation than the following illustration : 



3 4 61 = 12 

288 

679 



111 
132 

876 



(33) 



10. If each constituent of a row or column can be expressed as 
the sum or difference of two or more terms, the determinant can be 
expressed as the sum or difference of two other determinants. 

This can be proved by expanding each of the following determinants, and 
rearranging the letters. 

la l p, b lt ^1 = 10, b, c^lp b, Cl L . (34) 

In general, if each constituent of a row or column consists of n terms, the 
determinant can be expressed as the sum of n determinants. 

EXAMPLE. Show by this theorem, that 

16 + c, a - 6, a I = 3abc - a 3 - 6 3 - c". 
c + a, b - c, b\ 
a + b, c - a, c\ 

11. The value of a determinant is not changed by adding to or 
subtracting the constituents of any row from the corresponding 
constituents of one or more of the other rows or columns. 

Thus from 10 and 3, 



i t 

2 




(35) 



which proves the rule, because the determinant on the right vanishes. This 
result is employed in simplifying determinants. 
EXAMPLES. (1) Show 1 1, x t y + z I = 0. 

1, y, z + *\ 

1, z, x + y I 



410 



HIGHER MATHEMATICS. 



168. 



Add the second column to the last and divide the last column by x + y + z. 
The determinant vanishes (3). 



(2) Show 


x y z 


= (x + y + z) 


111. 




z x y 




z x y 




y z x 




y z x 


Add the second and third rows to the first and divide by x + 


(3) Why is a : 6 X 


<*i 


not equal to 


ftj + 6j, Oj + ftj, Cj 


a 2 b. 2 


% 




2 + & 2 , & 2 + 2> C 2 


a 3 6 3 


c s 




#3 + & 3 , 63 + &g C 3 


(4) Show 


4 1 


7 


= - 23. 




3 6 - 


- 2 






5 1 


8 





+ z. 



12. If all but one of the constituents of a row or column are 
cyphers, the determinant can be reduced to the product of the one 
constituent, not zero, into a determinant whose order is one less than 
the original determinant. 



For example, 
II a b 
aj 6 : 
6 




5 

- 3 



0-7 
6-2 

1 8 



- 7 



(36) 



The converse proposition holds. The order of a determinant can be raised 
by similar and obvious transformations. 

13. If all the constituents of a determinant on one side of the 
diagonal from the top left-hand corner are cyphers, the determinant 
reduces to the leading term. 

Thus I a a 6 X c x I = a a I 6 2 c 2 I = Ojbfa (37) 

6 2 c 2 lo cj 

10 cj 

The determinant I 6 2 c 2 I is called the co-factor or complement of the consti- 
tuent a,. c, 



168. The Multiplication of Determinants. 

This is done in the following manner : 

= \ a i d i + Vi> a i d -2 + b i 



( 38 ) 



The proof follows directly on expanding the right side of the 
equation. We thus obtain, 
= I ai d lt V 2 + 

a 2 d lt b 2 e 2 



a., 



h b 2 



= \a l b^ 
\a. 2 b. 2 



d l 



$169. DETERMINANTS. 411 

Since the value of a determinant is not altered by writing the 
columns in rows and the rows in columns, the product of two 
determinants may be written in several equivalent forms which all 
give the same result on expansion. Thus, instead of the right 
side of (38), we may have 



-f b. 2 e., b^ + b.,d. 2 , b^ + b. 2 e 2 

EXAMPLES. (1) Multiply I a l b l c 1 1 and 
l2 b i C 2\ 
\<h b s C J J e 3 fz 

The answer may be written in several different forms ; one form is 

I Mi + Vi'+ c i/i. M 2 + V 2 + c i/ 2 > Ms + Vs + c i/s 
Ml + Vl + C 2/l. 2^2 + &^2 + C 2/2' M 3 + V 3 + C 2/3 
Ml + Vl + C 3/l 3 rf 2 + V 2 + C 3/2. 5^3 + V 3 + C 3/3 

This can be verified by the laborious operation of expansion. There are 
twenty-seven determinants all but six of which vanish. 
(2) K b, 2 = I a\ + b\, a^+b 



When two constituents of a determinant hold the same relative position 
with respect to the rows and columns, they are said to be conjugate. Thus 
in the last of the determinants in (34) 6 X and q are conjugate, so are 6 3 and c 2 , 
r and Cj. If the conjugate elements are equal, the determinant is symmetrical, 
if equal but opposite in sign, we have a skew determinant. The square of a 
determinant is a symmetrical determinant. 



169. The Differentiation of Determinants. 

Suppose that the constituents of a determinant are independent 
and that 

D = 



then, d(D) = x^dy., + yjlx^ - x. 2 dy l - 

= (yJXi - y^dx^ + (x^y^ - xdy^ ; 

= I dx 1 y l I + I x l dy l I. . . (39) 

I ^2 y-2 1 I x -2 dy* I 

If the constituents of the determinant are functions of an in- 
dependent variable, say t, then, writing ^ for dx/dt, y, 2 for dy 2 /dt 
and so on, it can be proved, in the same way, 

D = \ Xl yi \, d(D)/dt = I ^ y, I -f I ^ y, I. . (40) 

I * 2 2/2! Us 2/2 



412 HIGHER MATHEMATICS. 

EXAMPLES. (1) Show that if D = I x l y l a l \; 



170, 



d(D) 



d(D)/dt = 



dx 1 



* 2 2/2 

\ X 3 2/3 

x l dy l 



+ I x l y l dz l 
\x 2 7/2 dz 2 
Ns 2/3 dz 3 






x l 



2/2 



(2) If a lt 6 2 , c lt a 2 , 6 2 , . . ., are constants, show that 

c i z I + > e ^ c -> dx 
a 2 x b 2 y c%z 



a 2 dx 



etc. 






170. Jacobians and Hessians. 

1. Definitions. If u, v, w, be functions of the independent 
variables, x, y, z, the determinant 

. . . . (41) 



is called a Jacobian and is variously written, 

(. *, ) . 



or 



or 



J > 



(42) 



when there can be no doubt as to the variables under consideration. 
In the special case, where the functions u, v, w are themselves 
differential coefficients of the one function, say u, with respect to 
x, y and z, the determinant 

. . . (43) 



is called a Hessian of u and written H(u), or simply H. The 
Hessian, be it observed, is a symmetrical determinant whose 
constituents are the second differential coefficients of u with 
respect to x, y, z. In other words, the Hessian of the primitive 
function u, is the Jacobian of the first differential coefficients of u, 
or in the notation of (42), 



$170. 



DKTKKMINANTS. 



H() = 



<)#' DV' &*/ 



. (44) 



2. Jacobians and Hessians of interdependent functions. If 
u = f(v), 



Eliminate the function f(v) as described on page 340. 



or, 



(45) 



That is to say, if u is a function of v, the Jacobian of the functions 
of u and v with respect to x and y will be zero. 

The converse of this proposition is also true. If the relation (45) 
holds good, u will be a function of v. 

In the same way, it can be shown that only when the Hessian 
of u is equal to zero, are the first derivatives of u with respect to x 
and y independent of each other. 

3. The Jacobian of a function of a function. If u v u 2 , are 
functions of x l and x. 2 , and x l and x 2 are functions of y l and y. 2 , 



By the rule for the multiplication of determinants, 




(46) 



or, 



This bears a close formal analogy with the well-known 



4. The Jacobian of implicit * functions. If u and v, instead of 

* A function is said to be explicit when it can be expressed directly in terms of 
the variable or variables, e.g., z is an explicit function of x in the expression : 2 = .c- ; 
z + a = bz*. A function is implicit when it cannot be so expressed in terms of the 
independent variable. Thus .f 2 + 2xy = y z ; x + y = z*, are implicit functions. 



414 



HIGHER MATHEMATICS. 



being explicitly connected with the independent variables x and y, 
are so related that 

P = fi( x * y,u,v) = Q; q = f 2 (x, y, u, v) = 0, 

u and v may be regarded as implicit functions of x and y. By 
differentiation 



^N * ^ i ^ ^ " > < ~r < ^ T ^ 



to 



0; 



to 



u(7 U f\ :L ~L v tv Q Ou >-w 

^v ^x ' to ^u ^y ^v ^x 



and by the rule for the multiplication of determinants, 

x 



to 



to to 
to ~dv 



Or, 



>,<?) 



(47) 



_ 

D(u, v) ' ?>(x, y) ^(x, y)' 
A result which may be extended to include any number of inde- 
pendent relations. 



171. Some Thermodynamic Relations. 

Determinants, Jacobians and Hessians are continually appearing in 
different branches of applied mathematics. 

The following summarises a paper by J. E. Trevor in the Journal of 
Physical Cliemistry (3, 523, 573, 1899). The results will serve as a simple 
exercise on the mathematical methods of some of the earlier sections of this 
work. The reader should find no difficulty in assigning a meaning to most 
of the coefficients considered. 

If U denotes the internal energy, < the entropy, p the pressure, v the 

volume, 6 the absolute temperature, Q the quantity of heat in a system of 

constant mass and composition, the two laws of thermodynamics state that 

dQ = dU + p . dv; dQ = 6d<t> (1) 

To find a value for each of the partial derivatives 



dv 



"dv* 



dv\ C&v\ /cH 

i \dpjj ( 

in terms of the derivatives of U. 

Case i. When v or <f> is constant. From (1), 

-p = 'dUI'dv ; and = 3 7/30 

First, differentiate each of the expressions (2), with respect to 
volume 



(2) 
at constant 



DETERMINANTS. n , 

By division, -(**}- ttillsS (4) 



Next, differentiate each of equations (2) with respect to v at constant entropy. 



By division, ~ ( ) = ^ gSc (6) 

Case ii. When either p or 6 is constant. We know that 

'dv 3<p 'dv 9</> 

First, when p is constant, eliminate dv or d<f> between equations (7). Hence 
show that 



where J denotes the Jacobian c)(p, 0)/B(v, </>). If H denotes the Hessian of E7, 
show that 



\dfjp "WU' \d6Jp~ H ' \de) P H ' 
Finally, if is constant, show that 



w ; \^)r "r- ; - \^)r ~ 

See also Baynes' Tliermodynamics (Oxford, 1878), pp. 95 et seq. 



416 



CHAPTEE XI. 
PROBABILITY AND THE THEORY OF ERRORS. 

172. Probability. 

" Perfect knowledge alone can give certainty, and in Nature perfect 
knowledge would be infinite knowledge, which is clearly beyond 
our capacities. We have, therefore, to content ourselves with 
partial knowledge knowledge mingled with ignorance, producing 
doubt." W. STANLEY JEVONS. 

" Lorsqu'il n'est pas en notre pouvoir de discerner les plus vraies 
opinions, nous devons suivre les plus probables." * RENE 
DESCARTES. 

NEARLY every inference we make with respect to any future event 
is more or less doubtful. If the circumstances are favourable, a 
forecast may be made with a greater degree of confidence than 
if the conditions are not so disposed. A prediction made in ignor- 
ance of the determining conditions is obviously less trustworthy 
than one based upon a more extensive knowledge. If a sports- 
man missed his bird moYe frequently than he hit, we could safely 
infer that in any future shot he would be more likely to miss than 
to hit. In the absence of any conventional standard of compari- 
son, we could convey no idea of the degree of the correctness of 
our judgment. The theory of probability seeks to determine the 
amount of reason which we may have to expect any event when 
we have not sufficient data to determine with certainty whether it 
will occur or not and when the data will admit of the application 
of mathematical methods. 

A great many practical people imagine that the "doctrine of 
probability " is too conjectural and indeterminate to be worthy of 



* Translated : " When it is not in our power to determine what is true, we ought 
to act according to what is most probable ". 



I 17-J. PROBABILITY AM) THE TIIKoRV OF KKKoi;-. 417 

serious study. Liagre* very rightly believes that this is due to 
the connotation of the word " probability". The term is so vague 
that it has undermined, so to speak, that confidence which we 
usually repose in the deductions of mathematics. So great, indeed, 
has been the dominion of this word over the mind that all applica- 
tions of this branch of mathematics are thought to be affected with 
the unpardonable sin want of reality. Change the title and the 
"theory" would not take long to cast off its conjectural character, 
and to take rank among the most interesting and useful applications 
of mathematics. 

Laplace remarks at the close of his Essai philosophique sur les 
Probabilites (Paris, 1812), " the theory of probabilities is nothing 
more than common-sense ( reduced to calculation. It determines 
with exactness what a well-balanced mind perceives by a kind 
of instinct, without being aware of the process. By its means 
nothing is left to chance either in the forming of an opinion, 
or in the recognising of the most advantageous view to select 
whenever the occasion should arise. It is, therefore, a most 
valuable supplement to the ignorance and frailty of the human 
mind. ..." 

1. If one of two possible events occurs in such a way that one of 
the events must occur in a ways, the other in b ways, the probability 
that the first will happen is a/ (a + b), and the probability that the 
second will happen is b/(a + b). 

If a rifleman hits the centre of a target about once every twelve shots 
under fixed conditions of light, wind, quality of powder, etc., we could say 
that the value of his chance of scoring a " bullseye " in any future shot is 1 in 
12, or T V, and of missing, 11 in 12, or f. If a more skilful shooter hits the 
centre about five times every twelve shots, his chance of success in any future 
shot would be 5 in 12, or T \, and of missing T 7 7 . 

Putting this idea into more general language, if an event can happen in 
a ways and fail in 6 ways, 

the probability of the event happening = a/ (a + b) ;} ,. 

the probability of the event failing = b/(a + 6),J ' 

provided that each of these ways is just as likely to happen as to fail. By 
definition, 

Number of ways the event occurs 

Jrrobability = . r2\ 

Number of possible ways the event may happen 

* Liagre 's Calcul des Probabilites (C. Muquardt, Bruxelles, 1879). 
f Literally " bons sens " = good sense. 
DD 



418 HIGHER MATHEMATICS. 172. 

2. If p denotes the probability that an event will happen, 1 - p 
denotes the probability that the event will fail. 

The shooter at the target is certain either to hit or to miss. In mathe- 
matics, unity is supposed to represent certainty, therefore, 

Probability of hitting + Probability of missing = Certainty = 1. (3) 

If the event is certain not to happen the probability of its occurrence is zero. 
Certainty is the unit of probability. Degrees of probability are fractions of 
certainty. 

Of course the above terms imply no quality of the event in itself, but 
simply the attitude of the computer's own mind with respect to the occurrence 
of a doubtful event. We call an event impossible when we cannot think of a 
single cause in favour of its occurrence, and certain when we cannot think of 
a single cause antagonistic to its occurrence. All the different " shades " of 
probability improbable, doubtful, probable lie between these extreme limits. 

Strictly speaking there is no such thing as chance in Nature. The irre- 
gular path described by a mote "dancing in a beam of sunlight" is determined 
as certainly as the orbit of a planet in the heavens. The terms "chance" and 
" probability " are nothing but conventional modes of expressing our ignor- 
ance of the causes of events as indicated by our inability to predict the results. 
" Pour une intelligence (omniscient)," says Liagre, " tout evenement a venir 
serait certain ou impossible.' 1 '' 

3. The probability that both of two independent events will 
happen together is the product of their separate probabilities. 

Let p denote the probability that one event will happen, q the probability 
that another event will happen, the probability that both events will happen 
together is 

pg. (*) 

This may be illustrated in the following manner: A vessel A contains 
a x white balls, b^ black balls, and a vessel B contains a? white balls and i 2 
black balls, the probability of drawing a white ball from A is p l = Oil(a l + bj, 
and from JB, p z = a 2 /(a 2 + 6 2 ). The total number of pairs of balls that can be 
formed from the total number of balls is (oj + 6 a ) (a 2 + & 2 ). In any simul- 
taneous drawing from each vessel, the probability that 

two white balls will occur is : a 1 a 2 /(a l + bj (a 2 + 6 2 ) ; . . . (5) 

two black balls will occur is : b l b. 2 j(a l + bj (a z + 6 2 ) ; . . . (6) 

white ball drawn first, black ball next, is : 0^(0^ + bj (a^ + 6 2 ) ; . (7) 

black ball drawn first, white ball next, is : a 2 6 1 /(c 1 + 6,) (04 + 6 2 ) " ( 8 ) 

black and white ball occur together, is : (ajb 9 + 6 1 o 2 )/(o 1 + 6 X ) (a,, + 6 2 ). (9) 

The sum of (5), (6), (9) is unity. This condition is required by the above 
definition. 

An event of this kind, produced by the composition of several events, is 
said to be a compound event. To throw three aces with three dice at one trial 
is a compound event dependent on the concurrence of three simple events. 



PROBABILITY AND THE THEORY OF ERRORS. 419 

Errors of observation are compound events produced by the concurrence of 
several independent errors. 

EXAMPLE. If the respective probabilities of the occurrence of each of n 
independent errors is P,, P 2 , . . ., P w , the probability of the occurrence of all 
together is P a P 2 . . . P. 

4. The probability of the occurrence of several events ivhich 
cannot occur together is the sum of the probabilities of their 
separate occurrences. 

If p, q, . . . denote the separate probabilities of different events, the 
probability that one of the events will happen is, 

= P + q + ........ (10) 

EXAMPLE. A bag contains 12 balls two of which are white, four black, 
six red, what is the probability that the first ball drawn will be a white, 
black, or a red one ? The probability that the ball will be white is , a 
black J, etc. The probability that the first ball drawn shall be a black or a 
white ball is . 

5. If p denotes the probability that an event will happen on a 
single trial, the probability that it will happen r times in n trials is 



The probability that the event will fail on any single trial is 1 - p ; the 
probability that it will fail every time is (1 - p) n . The probability that it 
will happen on the first trial and fail on the succeeding n - 1 trials is 
p(l - p) n - l . But the event is just as likely to happen on the 2nd, 3rd, . . . 
trials as on the first. Hence the probability that the event will happen just 
once in the n trials is 

np(l - pY - 1 ...... (12) 

The probability that the event will occur on the first two trials and fail on 
the succeeding n - 2 trials is p 2 (l - p) n ~ 2 . But the event is as likely to 
occur during the 1st and 3rd, 2nd and 4th, . . . trials. Hence the probability 
that it will occur just twice during the n trials is 



pY- 2 ..... (13) 

The probability that it will occur r times in n trials is, therefore, represented 
y formula (11). 

6. If p denotes the very small probability that an event will 
happen on a single trial, the probability that it will happen r times 
in a very great number (n) trials, is 



* The student may here find it necessary to read over 191. 



420 HIGHER MATHEMATICS. $ 172. 

From formula (11), however small p may be, by increasing the number of 
trials, we can make the probability that the event will happen at least once 
in n trials as great as we please. The probability that the event will fail 
every time in n trials is (1 - p) n , and if p be made small enough and n great 
enough, we can make (1 - p) n as small as we please.* If n is infinitely great 
and p infinitely small, we can write n = n-l = n-2 = . . . 



n(n - 1) 
(1 - p)* = 1 - np + 2 , V - 



(approx.) ; 



= e ~ n ? (approx.) ...... (15) 

(14) follows immediately from (11) and (15). This result is very important. 

EXAMPLES. (1) If n grains of wheat are scattered haphazard over a 
surface s units of area, show that the probability that a units of area will 
contain r grains of wheat is 

(an} r _ 

~vr e " 

Thus, n . ds/s represents the infinitely small probability that the small space 
ds contains a grain of wheat. If the selected space be a units of area, we 
may suppose each ds to be a trial, the number of trials will, therefore, be 
a/ds. Hence we must substitute an/s for np in (14) for the desired result. 

(2) Using the above notation and reasoning, show that the probability 
that an event will occur at least r times in n trials is 

p n + np n -l q + n ( n ^~ > p n ~ *g> + . . . + pr q n - r. . . . (16) 

Sometimes a natural process proves far too' complicated to 
admit of any simplification by means of "working hypotheses". 
The question naturally arises, can the observed sequence of events 
be reasonally attributed to the operation of a law of Nature or to 
chance ? 

For example, it is observed that the average of a large number of. readings 
of the barometer is greater at nine in th'e morning than at four in the after- 
noon ; Laplace (Theorie analytique des Probability, p. 49, 1820) asked whether 
this was to be ascribed to the operation of an unknown law of Nature or to 
chance? Again, Kirchhoff (Monatsberichte der Berliner Akademie, Oct., 1859) 
inquired if the coincidence between 70 spectral lines in iron vapour and in 
sunlight could reasonably be attributed to chance. He found that the prob- 
ability of a fortuitous coincidence was approximately as 1 : 1,000000,000000. 
Hence, he argued that there can be no reasonable doubt of the existence of iron 
in the sun. Michell (Phil. Trans., 57, 243, 1767 ; see also Kleiber, Phil. Mag. 
[5], 24, 439, 1887) has endeavoured to calculate if the number of star clusters 
is greater than what would be expected if the stars had been distributed 

* The reader should test this by substituting small numbers in place of p, and 
large ones for n. Use the binomial formula of 98. See the remarks on page 481, 



.< 173. PROBAI'.IU TV AND TIIK THK<|;Y OF KKi:i:>. 421 

haphazard over the heavens. Schuster (Proc. Ray. Soc., 31, 337, 1881) has 
tried to answer the question, is the number of harmonic relations in the 
spectral lines of iron greater than what a chance distribution would give ? 
Mallet (Phil. Trans,, 171, 1003, 1880) and Strutt (Phil. Mag. [6], 1, 311, 1901) 
have asked, do the atomic weights of the elements approximate as closely to 
whole numbers as can reasonably be accounted for by an accidental coinci- 
dence? In other words, are there common-sense grounds for believing the 
truth of Prout's law, that " the atomic weights of the other elements are exact 
multiples of that of hydrogen " ? 

The theory of probability does not pretend to furnish an in- 
fallible criterion for the discrimination of an accidental coincidence 
from the result of a determining cause. Certain conditions must 
be satisfied before any reliance can be placed upon its dictum. 
For example, a sufficiently large number of cases must be avail- 
able. Moreover, the theory is applied irrespective of any know- 
ledge to be derived from other sources which may or may not 
furnish corroborative evidence. Thus Kirchhoffs conclusion as to 
the probable existence of Fe in the sun was considerably strength- 
ened by the apparent relation between the brightness of the 
coincident lines in the two spectra. 

For details of the calculations, the reader must consult the original 
memoirs. Most of the calculations are based upon the analysis in Laplace's 
old but standard TJUorie (I.e.). An excellent r6sum& of this latter work will 
be found in the Encyclopaedia Metropolitana. 

The more fruitful applications of the theory of probability to natural 
processes has been in connection with the kinetic theory of gases and the 
" law " relating to errors of observation. 

173. Application to the Kinetic Theory of Gases.* 

The following illustrations are based, in the first instance, on a 
memoir by Clausius (Phil. Mag. [4], 17, 81, 1859). For further 
developments, Meyer's The Kinetic Theory of Gases (Longmans, 
Green & Co., 1899) may be consulted. 

1. To show that the probability that a single molecule, moving 
in a swarm of molecules at rest, ivill traverse a distance x without 
collision is 

P = e-*, . . . (17) 

* The purpose of the kinetic theory of gases is to explain the physical properties 
ot gases from the hypothesis that a gas consists of a great number of molecules in 
rapid motion. I select, here and in 181, a few dfdm-tious which directly refer to 
tin- theory of probability. 



422 HIGHER MATHEMATICS. 17o. 

where I denotes the probable value of the free path the molecule 
can travel without collision, and x/l denotes the ratio of the path 
actually traversed to the mean length of the free path. "Free 
path " is denned as the distance traversed by a molecule between 
two successive collisions. The "mean free path" is the average 
of a great number of free paths of a molecule. 

Consider any molecule moving under these conditions in a given direction. 
Let a denote the probability that the molecule will travel a path one unit 
long without collision, the probability that the molecule will travel a path 
two units long is a . a, or a 2 , and the probability that the molecule will travel 
a path x units long without collision is, from (4), 

P = a*, ...... (18) 

where a is a proper fraction. Its logarithm is therefore negative. (Why ?) 

If the molecules of the gas are stationary, the value of a is the same 
whatever the direction of motion of the single molecule. From (15), therefore, 

P = e - *P, 

where I = I/log . We can get a clear idea of the meaning of this formula by 
comparing it with (15). Supposing the traversing of unit path is reckoned a 
"trial," x in (17) then corresponds with n in (15). 1/Z in (17) replaces p in (15). 
1/Z, therefore, represents the probability that an event (collision) will happen 
during one trial. If I trials are made, a collision is certain to occur. This is 
virtually the definition of mean free path. 

2. To show that the length of the path which a molecule, moving 
amid a swarm of molecules at rest, can traverse without collision is 
probably 



where A. denotes the mean distance between any two neighbouring 
molecules, p the radius of the sphere of action corresponding to the 
distance apart of the molecules during a collision, TT is a constant 
with its usual signification. 

Let unit volume of the gas contain N molecules. Let this volume be 
divided into N small cubes each of which, on the average, contains only one 
molecule. Let A. denote the length of the edge of one of these little cubes. 
Only one molecule is contained in a cube of capacity A 3 . The area of a cross 
section through the centre of a sphere of radius p, is irp 2 , (12), page 491. If 
the moving molecule travels a distance A., the hemispherical anterior surface 
of the molecule passes through a cylindrical space of volume 7rp 2 A, (25), page 
492. Therefore, the probability that there is a molecule in the cylinder irp-\ 
is to 1 as irp 2 \ is to A 3 , that is to say, the probability that the molecule under 
consideration will collide with another as it passes over a path of length A. 



.55 173. PROBABILITY AND TI1K TIIKoliY ol KItRORS. 423 

is -JT^A : A :! . The probability that there will be no collision is 1 - ir^/A 2 . 
From (17), 



According to the kinetic theory, one fundamental property of gases is 
that the intermolecular spaces are very great in comparison with the di- 
mensions of the molecules, and, therefore, pV/A'- 4 is very small in comparison 
with unity. Hence also \fl is a small magnitude in comparison with unity. 
Expand e ~ A/ ' according to the exponential theorem (page 230), neglect terms 
involving the higher powers of A, and 

c ~ A /' = 1 - \jl ...... (21) 

From (20) and (21), 

A' -& 

I = 1 5 or > p = e A3 ..... (22) 

r* 

EXAMPLE. The behaviour of gases under pressure indicates that p is very 
much smaller than A. Hence show that "a molecule passes by many other 
molecules like itself before it collides with another ". Hint. From the first 
of equations (22), 

l:\ = A- : p 2 *-. 
Interpret the symbols. 

3. To show that (19) represents the mean value of the free path 
of n molecules moving under the same conditions as the solitary 
molecule just considered. 

Out of n molecules which travel with the same velocity in the same 
direction as the given molecule, ne~*l l will travel the distance x without 
collision, and ne ~ (* + < t *)n w jn travel the distance x + do. 1 without collision- 
Of the molecules which traverse the path a 1 , 

K* _ jr + < ^\ _/ _!f5\ n - 

e l -e l \=ne <(l-e *J = f l dx, 

of them will undergo collision in passing over the distance dx. The last 
transformation follows directly from (21). The sum of all the paths traversed 
by the molecules passing x and x + dx is 



Since each molecule must collide somewhere in passing between the limits 
x = and x = oo, the sum of all the possible paths traversed by the n molecules 
before collision is 



and the mean value of these n free paths is 



Integrate the indefinite integral as indicated on page 168. Therefore, from (4), 

<= 

represents the mean free path of these molecules moving with a uniform 
velocity. 



424 



HIGHER MATHEMATICS. 



S 173. 



EXAMPLES. (1) A molecule moving with a velocity u enters a space filled 
with n stationary molecules of a gas per unit volume, what is the probability 
that this molecule will collide with one of those at rest in unit time ? 

Use the above notation. The molecule travels the space u in unit time. 
In doing this, it meets with iriip^u molecules at rest. The probable number 
of collisions in unit time is, therefore, irnp 2 u, which represents the probability 
of a collision in unit time. 

(2) Show that the probable number of collisions made in unit time by a 
molecule travelling with a uniform velocity u, in a swarm of N molecules at 
rest, is 



What is the relation between this and the preceding result ? Note, 
Number of Collisions = u/l ; and N\ s 1. 

4. The number of collisions made in unit time by a molecule 
moving with uniform velocity in a direction which makes an angle 
with the direction of motion of a sivarm of molecules also moving 
with the same uniform velocity is probably 

-p-2% sin ^0. . . . (24) 

We must first determine the relative velocity of the molecules moving in 

a direction at an angle Q with the velocity of the molecule under consideration. 

One of the elementary propositions of mechanics is the parallelogram of 

velocities, which states that " if two component velocities are represented in 

direction and magnitude by two sides of a 
parallelogram drawn from a point, the re- 
sultant velocity is represented in direction 
and magnitude by the diagonal of the 
parallelogram drawn from that point ". 
The parallelepiped of velocities is an ex- 
tension of the preceding result into three 
dimensions. " If three component velocities 
are represented in direction and magnitude 
by the adjacent sides of a parallelepiped .r, 
T/, z (Fig. 126), drawn from a point, their 
resultant velocity will be represented by 
the diagonal of a parallelepiped drawn from 
that point." Conversely, if the velocity of the moving system is represented 
in magnitude and direction by the diagonal u (Fig. 126) of a parallelepiped, 
this can be resolved into three component velocities represented by three sides 
x, y, z of the parallelepiped drawn from a point. From 48, 

x = u cos 6 ; y = u sin . cos < ; z = u sin . sin <f>. . . (25) 
Let the three velocities represented by x, y, z, be respectively denoted by 
v i> % v s an< ^ I 6 * 5 u be represented by v. Hence, from Euclid i., 47, 




(26) 



* I7:J. PROBAIJII.I TV AND THE THEORY OF ERRORS. 425 

If one set of molecules moves with a uniform velocity whose components 
i relative to a given molecule moving with a uniform velocity whose 
components are x, y, z, then the relative component velocities of one molecule 
with respect to the other considered at rest, is 

V = (* - ^) 2 ; , = (y - yj* ; *>, 2 = (* - *,). 

From (26), _ 

.-. v = V(x - *,) + (y - y,f + (z - ,). . . . (27) 
If we choose the three coordinate axes so that the x-axis coincides with the 
direction of motion of the given molecule, we may substitute these values in 
(25), remembering that cosO = 1, sinO = 0, 

.-. .r = u; y = 0; z = ..... (28) 
Substitute (28) and (25) in (27), 



v = \/(u - u cos 0) 2 + u^irfe . cos 2 <j> 



u?cos 2 8 + 
since sin% + cos 2 x = 1. Similarly, and for the same reason, 

v = u \/2 - 2cos0 = u \/2(l - cosd), 
from page 500, 1 - cos x = 2(sin z) 2 , 

.-. v = 2usin%e ...... (29) 

Having found the relative velocity of the molecules, it follows directly 
from (23) and (29), that 

(Number of collisions) = ^ = P ~%u sin 0. 

5. The number of collisions encountered in unit time by a mole- 
cule moving in a swarm of molecules moving in all directions, is 

l U f. .... (30) 

Let u denote the velocity of the molecules, then the different motions 
can be resolved into three groups of motions according to the converse of the 
parallelepiped of velocities. Proceed as in the last illustration. 

The number of molecules (n) moving in a direction between 6 and 6 + dd 
is to the total number of molecules (N) in unit volume as 

n : N = 2*- sin ede : 47r ; . . . . (31) 

or n = ^-ZVsin 6d6. 

Since the angle can increase from to 180, the total number of collisions is 



To get the total number of collisions, it only remains to integrate for all 
directions of motion between and 180. Thus if A denotes the number of 
collisions, 

A = ^ Tain J*. sin *fr; 



or, = / "sin 2 ^ . cos 

*' Jo 



_ 4 

~ 3 A' 
by the method of integration on page 186. 



426 HIGHER MATHEMATICS. $ 174. 

EXAMPLE. Find the length of the free path of a molecule moving in a 
swarm of molecules moving in all directions, with a velocity u. Ansr. 

= tt/4 = fAV (32) 

For the hypothesis of uniform velocity see 181. 

6. Assuming that two unlike molecules combine during a colli- 
sion, the velocity of chemical reaction between two gases is 

^ = kNN', .... (33) 

where JV and N' are the number of molecules of each of the two 
gases respectively contained in unit volume of the mixed gases, 
dx denotes the number of molecules which combine in unit volume 
in the time dt ; k is a constant. 

Let the two gases be A and B. Let A. and \' respectively denote the 
distances between two neighbouring molecules of the same kind, then, as 
above, 

A T A 3 = N'\'* = 1 (34) 

Let p be the radius of the sphere of action, and suppose the molecules com- 
bine when the sphere of action of the two kinds of molecules approaches 
within 2p, it is required to find the rate of combination of the two gases. 

The probability that a B molecule will come within the sphere of action of 
an A molecule in unit time is uirp 2 l\ s \ by (23). Among the N' molecules of B, 

N'^udt ; or NN'irp 2 udt, .... (35) 

A 

by (34), combine in the time dt. But the number of molecules which combine 
in the time dt is - dN = - dN', or, from (35), 

dN = dN' = - NWvfudt. 

If dx represents the number of molecules which combine in unit volume in 
the time dt. 

dx = dN = dN' = vptiiNN'dt. 
Collecting together all the constants under the symbol k, 

dx/dt = kNN'. 

EXAMPLE. Show the relation between (33) and Wilhelmy's law of mass 
action. 

J. J. Thomson's memoir, " The Chemical Combination of Gases," PhiL 
Mag. [5], 18, 233, 1884, might now be read with profit. 

17$. Errors of Observation. 

If a number of experienced observers agree to test, indepen- 
dently, the accuracy of the mark etched round the neck of a litre 
flask with the greatest precision possible, the inevitable result 
would be that every measurement would be different. Thus, we 
might expect 

1-0003; 0-9991; 1-0007; 1-0002; 1-0001; 0-9998; 






< 174. ri;<>l',Ai;il.ri Y AM) I HK TIIKnUY OF ERRORS. 427 

Kxactly the same thing would occur if one observer, taking every 
known precaution to eliminate error, repeats a measurement a great 
number of times. These deviations become more pronounced the 
nearer the approach to the limits of accurate measurement. The 
discrepancies doubtless arise from various unknown and therefore 
uncontrolled sources of error. 

The irregular deviations of the measurements from, say, the 
arithmetical mean of all are called accidental errors. In the 
following discussion we shall call them "errors of observation" 
unless otherwise stated. 

The simplest as well as the most complex measurements are 
invariably accompanied by these fortuitous errors. Absolute 
agreement is itself an accidental coincidence. Stanley Jevons 
says, "it is one of the mosb embarrassing things we can meet 
when experimental results agree too closely ". Such agreement 
should at once excite a feeling of distrust. 

The observed relations between two variables, therefore, should 
not be represented by a point in space, rather by a circle around 
whose centre the different observations will be 
grouped (Fig. 127). Any particular observation 
will find a place somewhere within the circum- 
ference of the circle. The diagram (Fig. 127) 
suggests our old illustration, a rifleman aiming 
at the centre of a target. The rifleman may be 
likened to an observer ; the place where the 
bullet hits, to an observation ; the distance be- 
tween the centre and the place where the bullet hits the target 
resembles an error of observation. A shot at the centre of the 
target is thus an attempt to hit the centre, a scientific measure- 
ment is an attempt to hit the true value of the magnitude 
measured. 

The greater the radius of the circle (Fig. 127), the cruder and 
less accurate the measurements ; and, vice versd, the less the mea- 
surements are affected by errors of observation, the smaller will 
be the radius of the circle. In other words, the less the skill of 
the shooter, the larger will be the target required to record his 
attempts to hit the centre. 




428 



HIGHER MATHEMATICS. 
175. The "Law" of Errors.* 



These errors may be represented pictorially another way. 
Suppose we had obtained experimental results affected by the 
errors shown in the following table: 



Positive 
Deviations from 
Me?n between 


Number 
of Errors. 


Percentage 
Number 
of Errors. 


Negative 
Deviations from 
Mean between 


Number 
of Errors. 


Percentage 
Number 
of Errors. 


0-4 and 0*5 


10 


2 


0-4 and 0'5 


10 


2 


0-3 and 0'4 


20 


4 


0-3 and 0-4 


20 


4 


0-2 and 0-3 


40 


8 


0-2 and 0*3 40 


8 


0-1 and 0-2 80 


16 


0-1 and 0-2 


80 


16 


0-0 and 0-1 


100 


20 


0-0 and O'l 100 


20 



The above table shows that among 500 observations, 10 were 
affected with errors of magnitude between + 0*4 and + 0'5 ; 20, 
or 4 / , with errors between + 0'3 and + 0'4 ; . . . and 100 ob- 
servations, or 20 / , were affected with errors numerically less than 

0-1. This is an ideal case, but a 
sufficiently close approximation 
to reality for our present pur- 
pose. 

Plot, as ordinates, the num- 
bers in the third column with the 
corresponding means of the two 
limits in the first column as 
abscissae. A curve similar to 
n nPNPri (Fig. 128) will be the 
result. 

By a study of the last two dia- 
grams, we shall find that there 




FIG. 128. Probability Curve. 



is a regularity in the grouping of these irregular errors which, 
as a matter of fact, becomes more pronounced the greater the 

* Venn (Logic of Chance, 1896) calls this the "exponential law of errors," a lav:, 
because it expresses a physical fact relating to the frequency with which errors are 
found to present themselves in practice. The "method of least squares" is no more 
than a rule showing how the best representative value may be extracted from a set of 
experimental results. Poincare, in the preface to his Th'ermodynamique (Paris, 1892), 
quotes the laconic remark, "everybody firmly believes in it (the law of errors), because 
mathematicians imagine that it is a fact of observation, and observers that it is a 
theorem of mathematics ". 



< 175. PROBABILITY AND TIIK THKoRY (>\- KKItOKS. |-_v 

iiuinber of trials we take into consideration. Thus, it is found 
that 

1. Small errors are more frequent than large ones. 

2. Positive errors are as frequent as negative errors. 

3. Very large positive or negative errors do not occur. 

Any mathematical relation between an error (x) and the frequency, 
or rather the probability, of its occurrence (y), must satisfy these 
characteristics. When such a function, 

y-&*), 

is plotted, it must have a maximum ordinate corresponding with 
no error ; it must be symmetrical with respect to the ?/-axis, in 
order to satisfy the second condition ; and as x increases numeri- 
cally, y must decrease until, when x becomes very large, y must 
become vanishingly small. Such is the curve represented by the 
equation, 

y-fe-* v , . . . . (i) 

where h and k are constants.* The graph of this equation, called 
the probability curve, or curve of frequency, or curve of errors, 
is obtained by assigning arbitrary constant values to h and k and 
plotting a set of corresponding values of x and y in the usual way.-f 

To find a meaning for the constant k, put x = 0, then y = k t 
that is the maximum ordinate of the curve. If we agree to define 
an error as the deviation of each measurement from the arith- 
metical mean, k corresponds with those measurements which 
coincide with the mean itself, or are affected by no error at all. 
The height at which the curve cuts the ?/-axis (Fig. 129) represents 
the magnitude of the arithmetical mean ; k has nothing to do with 
the actual shape of the curve beyond increasing the length of the 
maximum ordinate as the accuracy of the observations increases. 

To find a meaning for the constant h, put k = 1, and plot 
corresponding values of x and y for x = J, + 1, + {!, + 2, . . . 
when h = 1, J, J, . . . In this way, it will be observed that 
although all the curves cut the ?/-axis at the same point, the 
greater the value of h, the steeper will be the curve in the 
neighbourhood of the central ordinate Oy. The physical signifi- 



* Several attempts by Gauss, Hagen, Herschel, Laplace, etc., have been made to 
prove this " law ". Adrain appears to have been the first to deduce the above formula 
on theoretical grounds. (1808.) 

f Use Table XXIIL, page 519, or \oge~ * 2 * 2 - - 



430 



HIGHER MATHEMATICS. 



S 175. 



cation of this is that the greater the magnitude of h, the more 
accurate the results and the less will be the magnitude of the 
deviation of individual measurements from the arithmetical mean 
of the whole set. Hence Gauss calls h the absolute " measure 

of precision". If the curves a, 
b, c (Fig. 129) retained their pre- 
sent shape while transposed to cut 
the 2/-axis at the same point, we 
should obtain a very good idea of 
the effect of h in the above function. 
We must now submit our 
empirical " law " to the test of 

experiment. Bessel has compared 
FIG. 129.-Probability Curves. the em)rg of observation in 470 

astronomical measurements made by Bradley with those which 
should occur according to the law of errors. The results of this 
comparison are shown in the following table taken from Encke's 
paper in the Berliner Astronomisches Jahrbuch for 1834, p. 249 
(Taylor's Scientific Memoirs, 2, 317, 1841) : 






X umber of Errors of each 


Magnitude of Error in 
Parts of a Second of 


Magnitude. 


Arc, between : 








Observed. 


Theory. 


and 0-1 


94 


95 


0-1 and 0-2 88 


89 


0-2 and 0-3 78 


78 


0-3 and 0-4 


58 


64 


0-4 and 0-5 


51 


50 


0-5 and 0-6 


36 


36 


0-6 and 0-7 26 


24 


0-7 and 0-8 14 


15 


0-8 and 0-9 


10 


9 


0-9 and 1-0 


7 


5 


above 1-0 


8 


5 



This is a remarkable verification of the above formula. There is 
this disagreement, while the theory provides for errors of any 
magnitude, however large, in practice, there is a limit above which 
no error will be found to occur, but read 187. 



Airy and Newcomb have also shown that the number and magnitude of 
the errors affecting extended series of observations are in fair accord with 



. PROBABILITY AND THE THEORY OF ERRORS. 431 

theory. But in every case, the number of large errors actually found is in 
excess of theory. To quote one more instance, Newcomb examined 684 

rations of the transit of Mercury. According to the "law" of errors, 
there should be 5 errors numerically greater than + 27". In reality, 49 sur- 
passed these limits. 

The theory assumes that the observations are all liable to the same 
errors, but differ in the accidental circumstances which give rise to the errors.* 
Equation (1) is by no means a perfect representation of the law of errors. 
The truth is more complex. The magnitude of the errors seems to depend, 
in some curious way, upon the number of observations. In an extended 
series of observations the errors may be arranged in groups. Each group has 
a different modulus of precision. This means that the modulus of precision 
is not constant throughout an extended series of observations. 

The probability curve represented by the formula 

j-to-w; 

may be considered a very fair graphic representation of the law 
connecting the probability of the occurrence of an error with its 
magnitude. 

176. The Probability Integral. 

Let X Q , x v x. 2 , . . . x be a series of errors in ascending order 
of magnitude from # to x. Let the differences between the 
successive values of x be equal. If x is an error, the probability 
of committing an error between X Q and x is the sum of the separate 
probabilities fc~*V, ke~ k ^ t . . ., (4), 172, or 



= fcv%-V. . (1) 

*o 

It' the summation sign is replaced by that of integration, we must 
let dx denote the successive intervals between any two limits 
x (} and x, thus 



Now it is certain that all the errors are included between the limits 
+ oo, and, since certainty is represented by unity, we have 

1 = A 

from page 269. Or, 

k = h . dx / v TT. .... (3) 

* Some observers' results seem more liable to these large errors than others, 
<lue, perhaps, to carelessness, or lapses of attention. Thomson and Tait (I.e.), I 
presume, would call the abnormally large errors "avoidable mistakes". 



432 HIGHER MATHEMATICS. 176. 

Substituting this value of k in the probability equation (1), pre- 
ceding section, we get the same relation expressed in another 
form, namely, 

h 22 
-j l *dx, .... (4) 

a result which represents the probability of errors of observation 
between the magnitudes x and dx. By this is meant the ratio : 

Number of errors between x and x + dx 
Total number of errors 

The symbols y and P are convenient abbreviations for this cumbrous 
phrase. For a large number of observations affected with accidental 
errors, the probability of an error of observation having a magnitude 
x, is, 



which is known as Gauss' law of errors. This result has the 
same meaning as y = ke~ hZx ^ of the preceding section. In (4), dx re- 
presents the interval, for any special case, between the successive 
values of x. For example, if a substance is weighed to the 
thousandth of a gram, dx = O'OOl, if in hundredths, dx = O'Ol, 
etc. The probability that there will be no error is 

h.dx/Jv; .... (6) 

the probability that there will be no error of the magnitude of a 
milligram is 

O-OOlfc/vC .... <7) 

The probability that an error will lie between any two limits 
X and x is 



The probability that an error will lie between the limits and x is 



which expresses the probability that an error will be numerically 
less than x. We may also put 

X -^d(hx), . . . (10) 

which is another way of writing the probability integral (8). In 
(8), the limits are X Q and x ; and in (9) and (10), + x. 






< 177. PROBABILITY AND THE TIIKnliY OF ERRORS. 188 

EXAMPLE. Find conditions which will make h in Gauss' equation a 
maximum. Hence deduce Legendre's principle of least squares : The most 
probable value for the observed quantities is that for ivhich the sum of the 
*qiuirt"x <>f flic individual errors is a minimum. That is to say, 

z 2 ! + x 2 2 + . . . + z 2 ,, = a minimum, . . . (11) 
where x l , u- 2 , . . ., x nj represents the errors respectively affecting the first, 
second, and the nth observations. 

To illustrate the reasonableness of the principle of least squares, we may 
revert to an old regulation of the Belgian army in which the individual scores 
of the riflemen were formed by adding up the distances of each man's shots 
from the centre of the target. The smallest sum won " le grand prix " of the 
regiment. It is not difficult to see that this rule is faulty. Suppose that one 
shooter scored a 1 and a 3 ; another shooter two 2's. It is obvious that the 
latter score shows better shooting than the former. 

The shots may deviate in any direction without affecting the score. Con- 
sequently, the magnitude of each deviation is proportional, not to the magni- 
tude of the straight line drawn from the place where the bullet hits to the 
centre of the target, but to the area of the circle described about the centre 
of the target with that line as radius. This means that it would be better 
to give the grand prize to the score which had a minimum sum of the squares 
of the distances of the shots from the centre of the target.* This is nothing 
but a graphic representation of the principle of least squares, formula (11). 
In this way, the two shooters quoted above would respectively score a 10 and 
an 8. 

177. The Best Representative Value for a Set of 
Observations. 

It is practically useless to define an error as the deviation of 
any measurement from the true result, because that definition 
would imply a knowledge which is the object of investigation. 
What then is an error ? Before we can answer this question, we 
must determine the most probable value of the quantity measured. 
The only available data, as we have just seen, are always as- 
sociated with the inevitable errors of observation. The measure- 
ments, in consequence, all disagree among themselves within 
certain limits. In spite of this fact, the investigator is called 
upon to state definitely what he considers to be the most probable 
value of the magnitude under investigation. Indeed, every chemical 
or physical constant in our textbooks is the best representative value 
of a more or less extended series of discordant observations. 

For instance, giant attempts have been made to find the exact 

* See properties of similar figures, 192. 
EE 



434 HIGHER MATHEMATICS. 177. 

length of a column of pure mercury of one square millimetre 
cross-sectional area which has a resistance of one ohm at 0C. 
The following numbers have been obtained : 



106-33 ; 106-31 



106-24 ; 



106-32 : 106-29 106-21 ; 

106-32 ; 106-27 106-19, 

centimetres (Everett's Illustrations of the C.G.S. System of Units, 
p. 176, 1891). There is no doubt that the true value of the re- 
quired constant lies somewhere between 106-19 and 106-33 ; but 
no reason is apparent why one particular value should be chosen 
in preference to another. The physicist, however, must select one 
number from the infinite number of possible values between the 
limits 106-19 and 106-33 cm. 

What is the best representative value of a set of discordant 
results? The arithmetical mean naturally suggests itself, and 
some mathematicians start from the axiom : " the arithmetical 
mean is the best representative value of a series of discrepant 
observations ". 

Various attempts, based upon the law of errors, have been made 
to show that the arithmetical mean is the best representative value 
of a number of observations made under the same conditions and 
all equally trustworthy. The proof rests upon the fact that the 
positive and negative deviations, being equally probable, will ulti- 
mately balance each other as shown in example (1).* 

EXAMPLES. (1) If a lt a 2 , . . ., a n are a series of observations, a their 
arithmetical mean, show that the algebraic sum of the residual errors is 

(! - a) + ( 2 - a) + . . . + (a n - a) = 0. . . . (1) 
Hint. By definition of arithmetical mean, 

a = a, + a 2 + . . . + a n . or> ^ = ^ + ^ + . . . + n . 

Distribute the n a's on the right-hand side so as to get (1), etc. 

* Hinrichs' The Absolute Atomic Weights of the Chemical Elements, published 
while the last proofs were under my hands, criticises the selection (and the selectors) 
of the arithmetical mean as the best representative value of a set of discordant obser- 
vations. The following exercises were suggested to me after reading pages 1-20 of 
that work. 

EXAMPLES. (1) What does the arithmetical mean of the weights of a large 
number of shillings in current circulation represent? 

(2) Point out the fallacy implied in the words: "if we cannot use the arith- 
metical mean of a large number of simple weighings of actual shillings as the true 
value of a (new) shilling, how dare we assume that the mean value of a few deter- 
minations of the atomic weight of a chemical element will give us its true value ? " 



< 177. PROBABILITY AND THE THEORY OF ERRORS. 435 

(2) Prove that the arithmetical mean makes the sum of the squares of 
the errors a minimum. Hint. See page 464. 

NOTE. When calculating the mean of a number of observations which 
agree to a certain number of digits, it is not necessary to perform the whole 
of the addition. For example, the mean of the above nine measurements is 
written 

106 + ('33 + -32 + -32 + -31 + -29 + -27 + -24 + -21 + -19) = 106-276. 

Edgeworth, "The Choice of Means," Phil. Mag., [5], 24, 268, 1887, and 
several articles on related subjects are to be found in the same journal between 
1883 and 1889. 

The best representative value of a constant interval. When the 
best representative value of a constant interval x in the expression 
y = a + nx (where n is a positive integer 1, 2 . . .) is to be de- 
termined from a series of measurements x. 2 - x v x 3 - x 2 , . . ., 
which vary a little from the desired value x, the arithmetical mean 
cannot be employed because it reduces to (x n - x^l(n - 1), the 
same as if the first and last term alone had been measured. In 
such cases it is usual to refer the results to the expression 

(n - 1) (x n - xj + (n - 3) (x n _ t - x 2 ) + . . . , 

n(ri* - 1) 

which has been obtained from the last of equations (4), 106, by 
putting 

^(x) -l + 2+...+n- \n(n + 1) ; 

2(^) = 12 + 2* + . . . + n* = in(n + l)(2w + 1) ; 

= x + x. + . . . x ; ^x = x + 2x. + . . . + nx. 



Such measurements might occur in finding the length of a rod at different 
temperatures, the oscillations of a galvanometer needle, the interval between 
the dust figures in Kundt's method for the velocity of sound in gases, the 
influence of CH 2 on the physical and chemical properties of homologous 
series, etc. 

EXAMPLES. (1) In a Kundt's experiment for the ratio of the specific 
heats of a gas, the dust figures were recorded in the laboratory notebook at 
30-7, 43-1, 55-6, 67'9, 80'1, 92'3, 104 '6, 116'9, 129-2, 141-7, 154-0, 166-1 centi- 
metres. What is the best representative value for the distance between the 
nodes ? Ansr. 12 - 3 cm. 

(2) The following numbers were obtained for the time of vibration, in 
seconds, of the "magnet bar" in Gauss and Weber's magnetometer in some 
experiments on terrestrial magnetism: 3-25; 9-90; 16-65; 23-35; 30-00; 36*65; 
43-30 ; 50-00 ; 56-70 ; 63-30 ; 69-80 ; 76'55 ; 83-30 ; 89-90 ; 96-65 ; 103-15 ; 109-80 ; 
116-65 ; 123-25 ; 129-95 ; 136-70 ; 143-35. Show that the period of vibration is 
6-707 seconds. 



436 HIGHER MATHEMATICS. $ 178. 

178. The Probable Error. 

Some observations deviate so little from the mean that we may 
consider that value to be a very close approximation to the truth, in 
other cases the arithmetical mean is worth very little. The ques- 
tion, therefore, to be settled is, what degree of confidence may we 
have in selecting this mean as the best representative value of a 
series of observations ? In other words, how good or how bad are 
the results ? 

We could employ Gauss' absolute measure of precision to answer 
this question. It is easy to show that the measure of precision of 
two series of observations is inversely as their accuracy. If the 
probabilities of an error x lt lying between and l lt and of an error 
o: 2 , between and 1 2 , are respectively 



PI 




= l r f 

V^J o 



it is evident that when the observations are worth an equal degree 
of confidence, P l = P. 2 . 

.-. l l h l = I. 2 h 2 ; or, l^ : 1. 2 = h. 2 : h^ 

or the measure of precision of two series of observations is in- 
versely as their accuracy. An error ^ will have the same degree 
of probability as an error 1. 2 when the measure of precision of the 
two series of observations is the same. 

For instance, if /^ = 4/z- 2 , P l = P'. 2 when 1. 2 = 4^, or four times 
the error will be committed in the second series with the same 
degree of probability as the single error in the first set. In other 
words, the second series of observations will be four times as 
accurate as the first. 

On account of certain difficulties in the application of this 
criterion, its use is mainly confined to theoretical discussions. 

One way of showing how nearly the arithmetical mean repre- 
sents all the observations, is to suppose all the errors arranged 
in their order of magnitude, irrespective of sign, and to select a 
quantity which will occupy a place midway between the extreme 
limits, so that the number of errors less than the assumed error is 
the same as those which exceed it. This is called the probable 
error (German " der wahrscheinliche Fehler "), not " the most 
probable error," nor "the most probable value of the actual 
error ". 



; 17s. PROBABILITY AND THE THEORY OF ERRORS. 437 

The probable error determines the degree of confidence we may 
have in using the mean as the best representative value of a series 
of observations. For instance, the atomic weight of oxygen is 
said to be 15-879 with a probable error 0-0003 (H = 1). This 
means that the arithmetical mean of a series of observations is 
15 > 879, and the probability is J (i.e., the odds are even) that the 
true atomic weight of oxygen lies between 15'8793 and 15*8787. 

Referring to Fig. 128, let the units be so chosen that the total area bounded 
by the curve and the #-axis is unity. If PM and P'M' are drawn at equal dis- 
tances from Oy in such a way that the area bounded by these lines, the curve, 
and the x-axis (shaded part in the figure), is equal to half the unit area, half 
the total observations will have errors numerically less than OM, that is, OM 
represents the probable error, PM its probability. 

The number of errors greater than the probable error is equal 
to the number of errors less than it. Any error selected at ran- 
dom is just as likely to be greater as less than the probable error. 
Hence, the probable error is the value of x in the integral 



page 432. From Table X., page 514, when P = J, hx = 0-4769 ; 
or, if r is the probable error, 

hr = 0-4769 ..... (2) 
Now it has already been shown that 

y=*e-**, .... (3) 

VTT 

From page 418, therefore, the probability of the occurrence of the 
independent errors x lt x.^ . . ., x n is the product of their separate 
probabilities, or 

P -*,-** ... (4) 



For any set of observations in which the measurements have been 
as accurate as possible, h has a maximum value. Differentiating 
the last equation in the usual way, and equating dP/dh to zero, 



Substitute this in (2), 

r = + 0-6745 



But ^(x 2 ) is the sum of the squares of the true errors. The true 
errors are unknown. By the principle of least squares, when 



438 HIGHER MATHEMATICS. $ 178. 

measurements have an equal degree of confidence, the most prob- 
able value of the observed quantities are those which render the 
sum of the squares of the deviations of each observation from the 
mean, a minimum. Let ^(v 2 ) denote the sum of the squares of 
the deviations of each observation from the mean. If n is large, 
we may put 

2(O = 2(O; 
but if n is a limited number, 

2(i> 2 ) < 2(* 2 ), 

.-. S (x 2 ) = 2(v 2 ) + u 2 . . . (7) 

All we know about u 2 is that its value decreases as n increases, 
and increases when 2<(x 2 ) increases. It is generally supposed that 
the best approximation to u 2 oc {^(x 2 )}/n, is to write 



n n n - 1* 

(Compare u 2 with m 2 in the next section, 179.) Hence, 



r = 0-6747 , ... (8) 

\ n JL 

which is virtually Bessel's formula for the probable error of a single 
observation. 2(v 2 ) denotes the sum of the squares of the numbers 
formed by subtracting each measurement from the arithmetical 
mean of the whole series, n denotes the number of measurements 
actually taken. 

The probable error of the arithmetical mean of the whole series 
of observations is 

= 0-6745 J-^L ... (9) 

\ n(n - 1) 

The derivation of this formula is given as an exercise at the end of 
179. 

The last two results show that the probable error is diminished 
by increasing the number of observations. 

(8) and (9) are only approximations. They have no significa- 
tion when the number of observations is small. Hence we may 
write | instead of 0-6745. For numerical applications, see next 
section. 

The great labour involved in the squaring of the residual errors of a large 
number of observations may be avoided by the use of Peter's approximation 
formula. According to this, the probable error of a single observation is 

r = 0-8453 ,^tJL, .... (10) 

\'n(n - 1) 



$ 17<). PROBABILITY AND THE THKnKY <H BBBOBS, 130 

where 2( + v) denotes the sum of the deviations of every observation from the 
mean, their sign being disregarded. The probable error of the arithmetical 
mean of the whole series of observations is 



179. Mean and Average Errors. 

The arbitrary choice of the probable error for comparing the 
errors which are committed with equal facility in different sets of 
observations, appears most natural because the probable error 
occupies the middle place in a series arranged according to order 
of magnitude so that the number of errors less than the fictitious 
probable error, is the same as those which exceed it. There are 
other standards of comparison. In Germany, the favourite method 
is to employ the mean error (" der mittlere Fehler "), which is de- 
fined as the error whose square is the mean of the squares of all the 
errors, or the " error which, if it alone were assumed in all the 
observations indifferently, would give the same sum of the squares 
of the errors as that which actually exists ". 
We have seen in 176, (5), that the ratio, 

Number of errors between x and x + dx _ "> _ h '2 x z j 

Total number of errors " J^ 

Multiply both sides by a? 2 and we obtain 

Sum of squares of errors between x and x + dx _ " 2 _^27, 
Total number of errors ~~ \J^ X ( 

By integrating between the limits + GO and - GO we get 

Sum of squares of all the errors _ ~2,(X ) ^ fa /* * . 2 - fcV-j ,, 

TotaFsum of errors n ' J^J _ x 

Let m denote the mean error, 



For the integration, see 108. 

.-. r = 0-6745w. . . (2) 

From (8) and (9) preceding section, the mean error which affects 
each single observation is given by the expression 

. (3) 

and the mean error which affects the whole series of results, 



440 HIGHER MATHEMATICS. $ 179. 

The mean error must not be confused with the " mean of the 
errors," or, as it is sometimes called, the average error,* another 
standard of comparison denned as the mean of all the errors re- 
gardless of sign. If a denotes the average error, 

v) fc2 2-, 

-=7= xe~ hx dx = , = ; r = O84o3a. (5) 



The average error measures the average deviation of each 
observation from the mean of the whole series. It is a more 
useful standard of comparison than the probable error when the 
attention is directed to the relative accuracy of the individual 
observations in different series of observations. 

The average error depends not only upon the proportion in 
which the errors of different magnitudes occur, but also on the 
magnitude of the individual errors. The average error furnishes 
useful information even when the presence of (unknown) constant 
errors ( 182) renders a further application of the " theory of 
errors " of questionable utility, because it will allow us to com- 
pare the magnitude of the constant errors affecting different series 
of observations, and so lead to their discovery and elimination (see 
182). 

A COMMON FALLACY. The way some investigators refer to the smallness 
of the probable error affecting their results conveys the impression that this 
canon has been employed to emphasise the accuracy of the work. As a 
matter of fact, the probable error does not refer to tlie accuracy of the irork 
nor to the magnitude of the errors, but only to the proportion in which the 
errors of different magnitudes occur. Cf. page 467. 

The reader will be able to show presently that the average error 
(A) affecting the mean of n observations is given by the expression 



This determines the effect of the average error of the individual 
observations upon the mean, and serves as a standard for comparing 
the relative accuracy of the means of different series of experiments 
made under similar conditions. 

EXAMPLES. Tables VI., VII., VIII., IX., will be found to save a great 
deal of labour in calculating the probable and mean errors of a series of 
observations. 

* Some writers call our "average error" the " mean error," and our " mean error " 
the "error of mean square ". 



17'.. PROBABILITY AND THK THKoKY <>K KIMJnKs. 441 



(1) The following galvanometer deflections were obtained in some obser- 
vations on the resistance of a circuit: 87 -0, 36-8, 36-8, 86-9, 37'1. Find the 
probable and mean errors. This small number of observations is employed 
simply to illustrate the method of using the above formulae. In practical 
work, mean or probable errors deduced from so small a number of observations 
are of little value. 

Arrange the following table : 



N limber of 
Observation. 


Deflection 
Observed. 


Departure from 
Mean. 


f2. 


1 
2 
3 
4 
5 


37-0 
36-8 
36-8 
36-9 
37-1 


+ 0-08 
- 0-12 
- 0-12 
- 0-02 
+ 0-18 


0-0064 
0-0144 
0-0144 
0-0004 
0-0324 


Mean = 36-92 ; 2(v 2 ) = 0-0680. 



The numbers in the last two columns have been calculated from those in the 
second. 

Since n = 5, and writing for 0-6745, 

. Mean error of a single result = \ / 0'068/4 = 0-0041. 
Mean error of the mean = v'O'068/5 x 4 = 0'0058. 

Probable error of a single result = f x/0'068/4 = 0-0027. 
Probable error of the mean = I v/0'068/5 x 4 = 0-0035. 

Average error of a single result = -52/5 = -104. 
Average error of tlie mean = + -52/5 <s/5 = -0465. 

The mean error of the arithmetical mean of the whole set of observations 
is written, 

36-92 0-0006 ; 
the probable error, 

36-92 0-0004. 
It is unnecessary to include more than two significant figures. 

(2) Rudberg (Pogg. Ann., 41, 271, 1837) found the coefficient of expansion 
o of dry air by different methods to be a x 100 = Q'3643, 0-3654, 0'3644, 0-3650, 
0-3653, 0-3636, 0-3651, 0-3643, 0-3643, 0-3645, 0-3646, 0-3662, 0-3840, 0-3902, 
0-3652. Required the probable and mean errors on the assumption that the 
results are worth an equal degree of confidence. 

(3) From example (3), page 135, show that the mean error is the abscissa 
of the point of inflection of the probability curve. For simplicity, put h = 1. 

(4) Cavendish has published the result of 29 determinations of the mean 
density of the earth (Phil. Trans., 88, 469, 1798) in which the first significant 
figure of all but one is 5 : 4-88 ; 5-50 ; -61 ; -07 ; -26 ; -55 ; -36 ; -29 ; -58 ; 
-65 ; -57 ; -53 ; -62 ; -29 ; -44 ; -34 ; -79 ; -10 ; -17 ; "39 ; -42 ; -47 ; -63 ; -34 ; -46 ; 
-K): -75; -68; -85. Verify the following results : Mean = 5-45; 2( + v) = 5-04 ; 
2(u 2 ) = 1-367 ; M = -041 ; m = -221 ; It = -0277 ; r = '149 ; a = '18 ; 
A = + -033. 



442 HIGHER MATHEMATICS. $ 17l>- 

The following results are convenient for reference : 

1. The mean (or probable error) of the sum of a number of ob- 
servations is equal to the square root of the sum of the squares of 
the mean (or probable) errors of each of the observations. 

Let a?!, # 2 , represent two independent measurements whose sum, or differ- 
ence combines to make a final result X, so that 

.Y = *! + .r 2 . 

Let the mean errors of x l and x 2 , be m^ and w 2 respectively. If M denotes 
the mean error in X, 

A' + M = (x l + Wj) + (x 2 m 2 ). 

.\ + M = + m li + m 2 . 

However we arrange the signs of M, m^, w 2 , in the last equation, we can only 
obtain, by squaring, one or other of the following expressions : 

M 2 = m^ -\- 2mjm z + w 2 2 ; or, M 2 = w x 2 - 2m 1 m 2 + m 2 2 , 

it makes no difference which. Hence the mean error is to be found by taking 
the mean of both these results. That is to say, 

M* = m^ + 7 2 2 ; or, M = Jm* + w 2 2 , 

because the terms containing + in^n^ and - m l m 2 cancel each other. This 
means that the products of any pair of residual errors (W]W 2 , vn^ni^ . . .) in an 
extended series of observations will have positive as often as negative signs. 
Consequently, the influence of these terms on the mean value will be negligibly 
small in comparison with the terms m-f, m 2 2 , ?n 3 2 , . . ., which are always posi- 
tive. Hence, for any number of observations, 

M* = m* + m* + . . . ; or, M = V /K 2 + w 2 2 +...). . (7) 
From equation (2), page 439, the mean error is proportional to the probable 
error R, m-^ to r 1? . . ., hence, 

B = * + r, + . . v . ; or, fl = V^ 2 + r 2 2 +...).. . (8) 

In othe"r words, the probable error of the SUM or DIFFERENCE of 
two quantities A and B respectively affected with probable errors 
a and + b is 

E = v/a 2 + R . . . . (9) 

EXAMPLES. (1) The molecular weight of titanium chloride (TiCl 4 ) is 
known to be 188-545 '0092, and the atomic weight of chlorine 35-179 '0048, 
what is the atomic weight of titanium ? Ansr. 47*829 + -0213. Hints. 
188-545 - 4 x 35-179 = 47'829 ; E = v/(-0092) 2 + (4 x -0048) 2 = '0213. 

(2) The mean errors affecting 6 l and 2 i n the formula R = 7c(0 2 - 0^ are 
respectively + -0003 and + -0004, what is the mean error affecting 6. 2 - 6 l and 
3(0 2 - 0j) ? Ansr. -0005 and 0015. 

2. The probable error of the PRODUCT of two quantities A and 
B respectively affected with the probable errors a and b is 



R = (Ab)' 2 + (5a) 2 . . . . (10) 



$ 170. PROBABILITY AND I H K THKoKY <>F KRK()K> H:; 

EXAMPLES. (1) Thorpe found that the molecular ratio 
Ag : TiCl 4 = 100 : 44-017 -0031. 

Hence determine the molecular weight of titanium tetrachloride, given the 
atomic weight of silver = 107-108 -0031. Ansr. 188-583 -0144. Hint. 

R = V{( 4 x 107-108 x -0031) a + (44-017 x 4 x -0031) 2 }. 
(2) The specific heat of tin is -0537 with a mean error of + -0014, and the 
atomic weight of the same metal is 118-150 + -0089, show that the mean error 
the product of these two quantities (Dulong and Petit's law) is 6-38 -1654. 
If a third mean, C, with a probable error, + c, is included, 

R = \'(BCa)* + (ACby + (A Be)*. . . . (11) 



3. The probable error of the QUOTIENT (B -f- A) of two quantities 
A and B respectively affected ivith the probable errors + a and b is 



/7Bo 

Vbr 

^ 



+b 

R = -- (12) 



EXAMPLES. (1) It is known that the atomic ratio 

Cu : 2Ag = 100 : 339-411 -0039, 
what is the atomic weight of copper on the assumption that 

Ag = 107-108 -0031 ? 
Ansr. 63-114 + -0020. Hint. 



214-216 x -0039\ a 
339:411 1 + (-0062)2 H- 339-411 = + -0020. 

Cu : 2 x 107-108 = 100 : 339-411 ; .-. Cu = 63-114. 

(2) Suppose that the maximum pressure of the aqueous vapour (/ 2 ) in the 
atmosphere at 16 is found to be 8-2, with a mean error + -0024, and the 
maximum pressure of aqueous vapour (/ a ) at the dewpoint, at 16, is 13'5, 
with a mean error of + -0012. The relative humidity (li) of the air is given 
by the expression h = /!// 2 , Show that the relative humidity at 16 is 
'6074 -0022. 

4. The probable error of the PROPORTION 

A : B = C : x, 

where A, B, C, are quantities respectively affected with the probable 
errors a, + b, + c, is 



IfBCa 

Vbr 



, T (06)* + (Bey 

-, - . (13) 



EXAMPLE. Stas found that A gCl0 3 furnished 25-080 -0010 / of oxygen 
and 74-920+ -0003 / otAgCL If the atomic weight of oxygen is 15-879 -0003, 
what is the molecular weight of AgCl ? Ansr. 142-303 -0066. Hints. 
25-080 : 74-920 = 3 x 15-879 : x ; .-. x = 142-303. 



444 HIGHER MATHEMATICS. $ 170. 

If A: B = C + x: D + x, 



EXAMPLE. Stas found that 31-488 + -0006 grams of NHCl were equiva- 
lent to 100 grams of AgNOf,. Hence determine the atomic weight of nitrogen, 
given Ag = 107-108 -0031 ; Cl = 35-179 -0048 ; H = 1 ; O 3 = 47'637 -0009. 
Ansr. 13-911 -0048. 

5. The probable error of the arithmetical mean of a series of 
observations is inversely as the square root of their number. 

Let r lt r 2 , . . ., r n be the probable errors of a series of independent obser- 
vations a lt 2 , . . ., a n , which have to be combined so as to make up a final 
result u. Let the probable errors be respectively proportional to the actual 
errors da v da 2 , . . ., da n . The final result u is a function such that 

u=f(a lt Og, . . .,). 

The influence of each separate variable on the final result may be determined 
by partial differentiation so that 

du = do! + -dos + ...,. . . . (15) 



where da v da. 2 , . . . represent the actual errors committed in measuring %, 
2 , . . . ; the partial differential coefficients determine the effect of these 
variables upon the final result u ; and du represents the actual error in u 
due to the joint occurrence of the errors da^ da 2 , . . . 

If we employ B in place of du, i\ in place of da^, etc., square (15) and 
show that 



The arithmetical mean of n observations is 

u = (% + a 2 + . . . + 

therefore, 



n 2 
But the observations have an equal degree of precision, and therefore, 



This result shows how easy it is to overrate the effect of multi- 
plying observations. If R denotes the probable error of the mean 
of 8 observations, four times as many, or 32 observations must be 
made to give a probable error of ^E ; nine times as many, or 72 
observations must be made to reduce E to R, etc. 

EXAMPLES. (1) Two series of determinations of the atomic weight of oxygen 
by a certain process gave respectively 15*8726 -00058 and 15-8769 -00058. 
Hence show that the atomic weight is accordingly written 15-87475 -00041. 



< 180. PROBABILITY AND HIM TMKORY OF KRROH- 445 

(2) In the preceding section, 178, given formula (8) deduce (9). Hint. 
Use (17), present section. 

(8) Deduce Peter's approximation formulae (10) and (11), 3 178. Hint. 
Since 

2(* 2 )/n = S(i>)/( - 1), 
page 438, we may suppose that on the average 

2(x) : v/fi = 2(r) : \ f n - 1, 
etc. 

(4) Show that when n is large, the result of dividing the mean of the 
squares of the errors by the square of the mean of the errors is constant. 
Hint. Show that 



This has been proposed as .<i test of the fidelity of the observations, and of the 
accuracy of the arithmetical work. For instance, the numbers quoted in the 
example on page 468 give 2(v) = 55'53; 2(y 2 ) = 354-35 ; 7i = 14; constant = 1-60. 
The canon does not usually work very well with a small number of observations. 
(5) Show that the probable (or mean) error is inversely proportional to 
the absolute measure of precision. Hint. From (1) and (2) 

1 

r = r x constant, .... (19) 

etc. See $ 190. 



180. Numerical Values of the Probability Integrals. 

We have discussed the two questions : 

1. What is the best representative value of a series of measure- 
ments affected with errors of observations ? 

2. How nearly does the arithmetical mean represent all of a 
given set of measurements affected with errors of observation ? 

It now remains to inquire 

3. How closely does the arithmetical mean approximate to the 
absolute truth ? 

To illustrate, we may use the results of Crookes' model research 
on the atomic weight of thallium (Phil. Tram., 163, 277, 1874) : 

203-628; 203-632; 203-636; 203-638; 203-639; 



203-642; 203-644; 203-649; 203-650; 203-666" ^ Mean: 

The arithmetical mean is only one of an infinite number of possible 
values of the atomic weight of thallium between the extreme limits 
203-628 arid 203*666. It is very probable that 203*642 is not the 
true value, but it is also very probable that 203*642 is very near 
to the true value sought. The question " How near?" cannot be 
answered. Alter the question to " What is the probability that 



446 HIGHER MATHEMATICS. 180. 

the truth is comprised between the limits 203 '642 + x?" and the 
answer may be readily obtained however small we choose to make 
the number x. 

First, suppose that the absolute measure of precision (h) of the 
arithmetical mean is known. 

Table X. gives the numerical values of the probability integral 

"hjt 

e~ h ^d(hx), 



where P denotes the probability that an error of observation will 
have a positive or negative value equal to or less than x, h is the 
measure of the degree of precision of the results. 

When h is unity, the value of P is read off from the table 
directly. To illustrate, we read that when x = + O'l P = -112 ; 
when x = 0'2 P = '223 ; . . ., meaning that if 1,000 errors are 
committed in a set of observations with a modulus of precision 
h = 1, 112 of the errors will lie between + 0-1 and - O'l , 223 
between + 0'2 and - 0'2, etc. Or, 888 of the errors will exceed 
the limits O'l ; 777 errors will exceed the limits 0'2 ; . . . 

When h is not unity, we must use ~-j- t -r- , . . ., in place of 
0-1, 0-2, . . . 

EXAMPLES. (1) If hx = 0-64, P, from the table, is 0-6346. Hence 0-6346 
denotes the probability that the error x will be less than 0'64//i, that is to 
say, 63-46 / of the errors will lie between the limits + 0'64//i. The remaining 
36-54% will lie outside these limits. 

(2) Required the probability that an error will be comprised between the 
limits 0-3 (h = 1). Ansr. -329. 

(3) Required the probability that an error will lie between - O'Ol and 
+ 0-1 of say a milligram. This is the sum of the probabilities of the limits 
from to - 0-01 and from to + 0-1 (h = 1). Ansr. -0113 + -1125 = -1237. 

(4) Required the probability that an error will lie between +1-0 and +0-01. 
This is the difference of the probabilities of errors between 1-0 and zero and be- 
tween 0-01 and zero (h = 1). Ansr. -8427 - -0113 = -8314. 

This table, therefore, enables us to find the relation between the 
magnitude of an error and the frequency with which that error will 
be committed in making a large number of careful measurements. 
It is usually more convenient to work from the probable error R 
than from the modulus h. More practical illustrations have, in 
consequence, been included in the next set of examples. 






$ ISO. PROBABILITY AND TIIK TIIKoliV ()! KliRoRS 447 

Second, suppose that the probable error of the arithmetical mean 
is known. 

Table XI. gives the numerical values of the probability integral 

p _ 

where P denotes the probability that an error of observation of a 
positive or negative value, equal to or less than x, will be com- 
mitted in the arithmetical mean of a series of measurements with 
probable error r (or R). This table makes no reference to h. To 
illustrate its use, of 1,000 errors, 54 will be less than ^R ; 500 
less than R ; 823 less than 2R ; 957 less than 3R ; 993 less than 
4tR ; and one will be greater than 5R. 

EXAMPLES. (1) A series of results are represented by 6-9 with a probable 
error + 0-25. The probability that the probable error is less than 0-25 is . 
What is the probability that the actual error will be less than 0-75 ? Here 
xfR = 0-75/0-25 = 3. From the table, p = 0-9570 when x/R = 3. This means 
that 95-7 / of the errors will be less than 0'75 and 4-3 / greater. 

(2) Dumas has recorded the following 19 determinations of the chemical 
equivalent of hydrogen (O = 100) using sulphuric acid (H 2 S0 4 ) with some, and 
phosphorus pentoxide (P 2 O 5 ) as the drying agent in other cases : 

i. H Z S0 4 : 12-472, 12-480, 12-548, 12-489, 12-496, 12-522, 12-533, 12-546, 
12-550, 12-562; 

ii. P. 2 O 5 : 12-480, 12-491, 12-490, 12-490, 12-508, 12-547, 12-490, 12-551, 
12-551. Dumas' " Recherches sur la Composition de 1'Eau," Ann. de Chim. 
et de Phys. [3], 8, 200, 1843. 

What is the probability that there will be an error between the limits 
+ 0-015 in the mean (12-515), assuming that the results are free from constant 
errors ? The chemical student will perhaps see the relation of his answer to 
Prout's law. 

Hints, x/R = t R = -004685 ; x =? -015 ; .-. t = 3-2. From Table XI., 
when t = 3-2, P = -969. Hence the odds are 969 to 31 that the mean 12-515 
is affected by no greater error than is comprised within the limits + -015. 
To exemplify Table X., h = -4769/tf = 102, .-. hx = 102 x -015 = 1-53. From 
the table, P = -969 when hx = 1-53, etc. That is to say, 96-9 / of the errors 
will be less and 3-1 / greater than the assigned limits. 

(3) From Crookes' ten determinations of the atomic weight of thallium 
(above) calculate the probability that the atomic weight of thallium lies be- 
tween 203-632 and 203-652. Here x = 0-01 ; R = -0023 ; .-. t = x/R = 4-4. 
From Table XI., P = -997. (Note how near this number is to unity indicating 
certainty.) The chances are 332 to 1 that the true value of the atomic weight 
of thallium lies between 203-632 and 203-652. We get the same result by 
means of Table X. Thus h = -4769/-0023 = 207; .'. /w = 207 x -01 = 2-07. 
When hx = 2-07, P = -997. If 1,000 observations were made under the same 
conditions as Crookes', we could reasonably expect 997 of them to be affected 



448 HIGHER MATHEMATICS. $ 181. 

by errors numerically less than O'Ol, and only 3 observations would be affected 
by errors exceeding these limits. 

The rules and formulae deduced up to the present are by no 
means inviolable. The reader must constantly bear in mind the 
fundamental assumptions upon which we are working. If these 
conditions are not fulfilled, the conclusions may not only be super- 
fluous, but even erroneous. The necessary conditions are : 

1. Every observation is as likely to be in error as every other one. 

2. There is no perturbing influence to cause the results to have a 
bias or tendency to deviate more in some directions than in others. 

3. A large number of observations has been made. In practice, 
the number of observations may be considerably reduced if the 
second condition is fulfilled. In the ordinary course of things from 
10 to 25 is usually considered a sufficient number. 



181. Maxwell's Law of Distribution of Molecular Velocities. 

In a preceding discussion, the velocities of the molecules of a 
gas were assumed to be the same. Can this simplifying assump- 
tion be justified ? 

According to the kinetic theory, a gas is supposed to consist of 
a number of perfectly elastic spheres moving about in space with a 
certain velocity. In case of impact on the walls of the bounding 
vessel, the molecules are supposed to rebound according to known 
dynamical laws. This accounts for the pressure of a gas. 

The velocities of all the molecules of a gas in a state of equili- 
brium are not the same. Some move with a greater velocity than 
others. At one time a molecule may be moving with a great 
velocity, at another time, with a relatively slow speed. 

The attempt has been made to find a law governing the distri- 
bution of the velocities of the motions of the different molecules, 
and with some success. Maxwell's law is based upon the assump- 
tion that the same relations hold for the velocities of the molecules 
as for errors of observation. This assumption has played a most 
important part in the development of the kinetic theory of gases. 

The probability y that a molecule will have a velocity equal to 
x is given by an expression of the type : 



< isl. PROBABILITY AND THK TIIKOUY OF KRKOKS. ir 

A graphical representation of this law is readily obtained by plotting 
corresponding values of x and y in the usual way. 

Very few molecules will have velocities outside a certain restricted 
range. It is possible for a molecule to have any velocity whatever 
but the probability of the existence of velocities outside certain 
limits is vanishingly small. 

The reader will get a better idea of the distribution of the velocities of the 
molecules by plotting the graph of the above equation for himself. Remember 
that the ordinates are proportional to the number of molecules, abscissae to 
their speed. Areas bounded by the x-axis, the curve and certain ordinates 
will give an idea of the number of molecules possessing velocities between the 
abscissae corresponding to the boundary ordinates. Use Table XXIII. 

Returning to the study of the kinetic theory of gases, 173, 
the number of molecules with velocities between v and v + dv is 
assumed to be represented by an equation analogous to the ex- 
pression employed to represent the errors of mean square in 179, 
namely, 



where N represents the total number of molecules, a is a constant 
to be evaluated. 

1. To find a value for the constant a in terms of the average 
velocity ( F ) of the molecules. 

Since there are dN molecules with a velocity v, the sum of the velocities 
of all these dN molecules is vdN, and the sum of the velocities of all the 
molecules must be 



v.dN. 
= o 



From (2), 



4 r 

n =j= 
a'\W 



(How did N vanish ?) Hence, 

= *V x'. ..... (3) 

2. To find the average velocity of the molecules of a gas. 

By a well-known theorem in elementary mechanics, the kinetic energy of 
a mass m moving with a velocity v is tynv 2 . Hence, the sum of the kinetic 
energies of the dN molecules will be %(mdN)v z , because there are dN molecules 

FF 



450 HIGHER MATHEMATICS. 181. 

moving with a velocity v. From (2), therefore, the total kinetic energy (T) of 
all the molecules is 



/ = GO n Af m r x #2 

T= / lmv*.dN==^ F : t ,4 e ~a*dv 
.',= o-Vir.'o 

= %Nma? = f Afo 8 . 



(4) 

where If = .Nw = total mass of N molecules each of mass m. 
The total kinetic energy of N molecules of the same kind is 

T = \mv* + $mv 2 2 + . . . + fynvj? = %m(v + v 2 2 + . . - + vjp). (5) 
The velocity of mean square ( U] is defined as the velocity whose square 
is the average of the squares of the velocities of all the N molecules, or, 



From (5), therefore, 

T = mNU* = MU* ...... (6) 

From (4) and (6), therefore, O2> 

a = ^|; and F =^j= = "9213 C7. .... (7) 

Most works on chemical theory * give a simple method of proving that if 
p denotes the pressure and p the density of a gas, 

P = *pU* ....... (8) 

This in conjunction with (6) allows the average velocity of the molecules of a 
gas to be calculated from the known values of the pressure and density of the 
gas. 

NUMERICAL EXAMPLE. One c.c. of hydrogen gas weighs '0000896 grams 
under standard barometric pressure, 76 cm. of mercury. Specific gravity of 
mercury = 13 '5. Hence, a column of mercury 76 cm. long and 1 sq. cm. 
-cross-section weighs 76 x 13-5 = 1033-2 grams. But, 

Weight = Mass x Acceleration of gravity, 

Weight of unit volume 
p Density = Mass of unit volume = - - 

= -0000896/981 = -000009. 
From (7) and (8), 



= -9213 . -9213 = 184,000. 



That is to say, the average velocity of hydrogen molecules under atmospheric 
pressure at C. is approximately 184,000 centimetres per second. 

3. To show that the average velocity of the molecules of a gas is 
proportional to its rate of diffusion. 

This will be left as an exercise. Hint. Use (7) and (8) above, and (1), 
95. See also (2), 190. 

The reader is no doubt familiar with the principle underlying 
Maxwell's law, and, indeed, the whole kinetic theory of gases. I 

* E.g., Ramsay's Experimental Proofs of Chemical Theory for Beginners. 



L82. PROBABILITY AND T1IK THKORV <>l BRRORa i:,l 

nijiy mention two examples. The number of passengers on say 
thf 3.10 P.M. suburban daily train is fairly constant in spite of the 
fact that that train does not carry the same passengers two days 
running. Insurance companies can average the number of deaths 
per 1,000 of population with great exactness. Of course I say 
nothing of disturbing factors. A bank holiday may require pro- 
vision for a supra-normal traffic. An epidemic will run up the 
death rate of a community. The commercial success of these 
institutions is, however, sufficient testimony of the truth of the 
method of averages, otherwise called the statistical method 
of investigation. The same type of mathematical expression is 
required in each case. 

It will thus be seen that calculations, based on the sup- 
position that all the molecules possess equal velocities, are quite 
admissible in a first approximation. The net result of the 
"dance of the molecules" is a distribution of the different velo- 
cities among all the molecules, which is maintained with great 
exactness. 

G. H. Darwin has deduced values for the mean free path, etc., from the 
hypothesis that the molecules of the same gas are not all the same size. He 
has examined the consequences of the assumption that the sizes of the mole- 
cules are ranged according to a law like that governing the frequency of errors 
of observation. For this, see his memoir " On the mechanical conditions of a 
swarm of meteorites'" (Phil. Trans., 180, 1, 1889). 



182. Constant Errors. 

The irregular accidental errors hitherto discussed have this 
distinctive feature, they are just as likely to have a positive as a 
negative value. But there are errors which have not this character. 
If the barometer vacuum is imperfect, every reading will be too 
small ; if the glass bulb of a thermometer has contracted after 
graduation, the zero point rises in such a way as to falsify all 
subsequent readings ; if the points of suspension of the balance 
pans are at unequal distances from the centre of oscillation of the 
beam, the weighings will be inaccurate. A change of tempera- 
ture of 5 or 6 may easily cause an error of 0-2 to 1'0/ in an 
analysis, owing to the change in the volume of the standard 
solution. Such defective measurements are said to be affected 



452 HIGHER MATHEMATICS. 182. 

by constant errors.* By definition, constant errors are produced 
by well-defined causes which make the errors of observation pre- 
ponderate more in one direction than in another. Thus, some of 
Stas' determinations of the atomic weight of silver are affected by 
a constant error due to the occlusion of oxygen by metallic silver in 
the course of his work. 

One of the greatest trials of an investigator is to detect and 
if possible eliminate constant errors. This is usually done by 
modifymg the conditions under which the experiments are per- 
formed. Thus the magnitude is measured under different condi- 
tions, with different instruments, etc. It is assumed that even 
though each method or apparatus has its own specific constant 
error, all these constant errors taken collectively will have the 
character of accidental errors. To take a concrete illustration, 
faulty "sights" on a rifle may cause a constant deviation of the 
bullets in one direction ; the " sights " on another rifle may cause 
a constant "error" ( 174) in another direction, and so, as the 
number of rifles increases, the constant errors assume the character 
of accidental errors and thus, in the long run, tend to compensate 
each other. This is why Stas generally employed several different 
methods to determine his atomic weights. To quote one practical 
case, Stas made two sets of determinations of the numerical value 
of the ratio Ag\: KCl. In one set, four series of determinations 
were made with KCl prepared from four different sources in con- 
junction with one specimen of silver, and in the other set different 
series of experiments were made with silver prepared from different 
sources in conjunction with one sample of KCl.-f 

The calculation of an arithmetical mean is analogous to the 
process of guessing the centre of a target from the distribution of 
the " hits " (Fig. 127). If all the shots are affected by the same 
constant error, the centre, so estimated, will deviate from the true 
centre by an amount depending on the magnitude of the (presumably 



* Personal error. This is another type of constant error which depends on the 
personal qualities of the observer. Thus the differences in the judgments of the 
astronomers at the Greenwich Observatory as to the observed time of transit of a star 
and the assumed instant of its actual occurrence is said to vary from r J 1T to ^ of a 
second, and to remain fairly constant for the same observer. Some persistently read 
the burette a little high, others a little low. Vernier readings, analyses based on 
colorimetric tests (such as Nessler's ammonia process), etc., may be affected by 
personal errors. 

f Unfortunately the latter set was never completed. 



< is:',. PROBABILITY AND THE TIIKuliY <>l BBROB& IN 

unknown) constant error. If this magnitude can be subsequently 
determined, a simple arithmetical operation (addition or subtraction) 
will give the correct value. Thus Stas found that the amount of 
potassium chloride equivalent to 100 parts of silver, in one case, 
was as 

Ag : KCl = 100 : 69-1209. 

The KCl was subsequently found to contain '00259 / of silica. 
The chemical student will see that -00087 has consequently to be 
subtracted from 69-1209. Hence, 

AH : KCl = 100 : 69-11903. 

After Lord Kayleigh (Proc. Roy. Soc., 43, 356, 1888, or rather 
Agamennone in 1885) pointed out that the capacity of an exhausted 
glass globe is less than when the globe is full of gas, all measure- 
ments of the densities of gases involving the use of exhausted 
globes had to be corrected for shrinkage. Thus Regnault's ratio, 
1 : 15-9611, for the relative densities of hydrogen and oxygen was 
" corrected for shrinkage " to 1 : 15-9105. The proper numerical 
corrections for the constant errors of a thermometer are indicated 
on the well-known " Kew certificate," etc. 

If the mean error of each set of results differs, by an amount 
to be expected, from the mean errors of the different sets measured 
with the same instrument under the same conditions, no constant 
error is likely to be present. The different series of atomic weight de- 
terminations of the same chemical element, published by the same 
(perhaps excluding Stas) or by different observers, do not stand this 
test satisfactorily. Hence, Ostwald concludes that constant errors 
must have been present even though they have escaped the ex- 
perimenter's ken. 

EXAMPLE. Discuss the following: "Merely increasing the number of 
experiments, without varying the conditions or method of observation, 
diminishes the influence of accidental errors. It is, however, useless to 
multiply the number of observations beyond a certain limit. On the other 
hand, the greater the number and variety of the observations, the more 
complete will be the elimination of the effects of both constant and accidental 
errors." 

183. Proportional Errors. 

One of the greatest sources of error in scientific measurements 
occurs when the quantity cannot be measured directly. In such 
cases, two or more separate observations may have to be made on 



454 HIGHER MATHEMATICS. 183. 

different magnitudes. Each observation contributes some little 
inaccuracy to the final result. Thus Faraday has determined the 
thickness of gold leaf from the weight of a certain number of 
sheets. Foucault measures time, Le Chatelier measures tempera- 
ture in terms of an angular deviation. The determination of the 
rate of a chemical reaction often depends on a number of more or 
less troublesome analyses .* 

For this reason, among others, many chemists prefer the standard O = 16 
as the basis of their system of atomic weights. The atomic weights of most 
of the elements have been determined directly or indirectly with reference to 
oxygen. If H 1 be the basis, the atomic weights of most of the elements 
depend on the nature of the relation between oxygen and hydrogen a 
relation which has not yet been fixed in a satisfactory manner. The best de- 
terminations made since 1887 vary between H: O = l : 15-96 and H: = 1 : 15-87. 
If the former ratio be adopted, the atomic weights of antimony and uranium 
would be respectively 119*6 and 239-0 ; while if the latter ratio be employed, 
these units become respectively 118-9 and 237'7, a difference of one and two 
units ! It is, therefore, better to contrive that the atomic weights of the 
elements do not depend on the uncertainty of the ratio H : O, by adopting 
the basis : = 16. 

If the quantity to be determined is deduced by calculation from 
a measurement, Taylor's theorem furnishes a convenient means of 
criticising the conditions under which any proposed experiment is 
to be performed, and at the same time furnishes a valuable insight 
into the effect of an error in the measurement on the whole result. 

It is of the greatest importance that every investigator should 

* Indirect results are liable to another source of error. The formula employed 
may be so inexact that accurate measurements give but grossly approximate result*. 
For instance, a first approximation formula may have been employed when the 
accuracy of the observations required one more precise ; TT - *?- may have been put 
l n place of ir 3'14159 ; or the coefficient of expansion of a perfect gas has Vven 
applied to an imperfect gas. Such errors are called errors of method. 

There is a well-defined distinction between the approximate values of a physical 
constant, which are seldom known to more than three or four significant figures 
(see 189), and the approximate value of the incommensurables TT, e, \ ; 2, . . . which 
can be calculated to any desired degree of accuracy. If we use -v 2 - in place of 3 '1416 
for TT, the absolute error is greater than or equal to 3 '1426 3 '1416, and equal to or 
less than 3-1428 - 3*1416 ; that is, between '0012 and '0014. In scientific work we 
are not concerned with absolute errors although it is assumed that the proportional 
error is an approximate representation of the ratio of the absolute error to the true 
value of the magnitude. 

By the way, " > " is a convenient abbreviation used in place of the phrase " is 
greater than or equal to," and " <" is used in place of "is equal to or less than". 
See page 10. 



, PROBABILITY AND Till! THEORY OF ERI:H; 

have a clear idea of the different sources of error to which his 
results are liable in order to be able to discriminate between im- 
portant and unimportant sources of error, and to find just where 
the greatest attention must be paid in order to obtain the best 
results. 

Let y be the desired quantity to be calculated from a magnitude 
x which can be measured directly and is connected with y by the 
relation 

</=/(*) 

f(x) is always affected with some error dx which causes y to deviate 
from the truth by an amount dy. The error will then be 

dy = (y + dy) - y = f(x + dx) - f(x). 
dx is necessarily a small magnitude, therefore, by Taylor's theorem, 

f(x + dx) = f(x) + f(x) . dx + . . ., 
or, neglecting the higher orders of magnitude, 

dy = f'(x) . dx. 

The relation between the error and the total magnitude of y is 
dy f(x) . dx 

y = /(*) 

The ratio dy/y is called the proportional, relative, or fractional 
error,* that is to say, the ratio of the error involved in the whole 
process to the total quantity sought. 

Students often fail to understand why their results seem all 
wrong when the experiments have been carefully performed and 
the calculations correctly done. For instance, the molecular 
weight of a substance is known to be either 160, or some multiple 
of 160. To determine which, -380 (or w) grm. of the substance 
was added to 14*01 (or wj grms. of acetone boiling at ^ 1 (or 3 '50) 
on Beckmann's arbitrary scale, the temperature, in consequence, fell 
to O. y (or 3 -36) ; the molecular weight of the substance (M) is then 
represented by the known formula 

M = 1670 - ; or, J/ = 1670-^ - = 323, 

or approximately 2 x 160. Now assume that the temperature 
readings may be + 0'05 in error owing to convection currents, 
radiation and conduction of heat, etc. Let 0^ = 3 '55 and 0. 2 = 3 '31, 



100 . (.lyly is the percentage error. 



456 HIGHER MATHEMATICS. $ 183. 

This means that an error of ~ in the reading of the thermometer 
would give a result positively misleading. This example is by no 
means exaggerated. The simultaneous determination of the heat 
of fusion and of the specific heat of a solid by the solution of two 
simultaneous equations, and the determination of the latent heat 
of steam are specially liable to similar mistakes. A study of the 
reduction formula will show in every case that relatively small 
errors in the reading of the temperature are magnified into 
serious dimensions by the method used in the calculation of the 
final result. 

EXAMPLE. The radius of curvature (r) of a lens, is given by the formula 

r = afl(f - a). 

(See any textbook on optics for the meaning of the symbols.) Let the true 
values of / and a be respectively 20 and 15. Let / and a be liable to error to 
the extent of + -5, say, / is read 20-5 and a, 14-5. Then the true value of r is 
60, the observed value 51-2. Fractional error = 8'8/60. This means that an 
error of about 2-5 / in the determination of / and a may cause r to deviate 
15 / from the truth. 

The degree of accuracy of a measurement is determined by the 
magnitude of the proportional error. 

Magnitude of error 
= Total magnitude of quantity measured* 

If we knew that an astronomer had made an absolute error of 
100,000 miles in estimating the distance between the earth and 
the sun, and also that a physicist had made an absolute error of 

* ne ioooo.oooooo tn f a m il e m measuring the wave length of a 
^spectral line, we could form no idea of the relative accuracy of the 
two measurements in spite of the fact that the one error is the 
looo.oooo^o.ooooooth Part of the other. In the first measurement 
the error is about 1|0 1 00 - of the whole quantity measured, in the 
second case the error is about the same order of magnitude as the 
quantity measured. In the former case, therefore, the error is neg- 
ligibly small ; in the latter, the error renders the result nugatory. 
The following examples will serve to fix these ideas : 

EXAMPLES. (1) It is required to determine the capacity of a sphere from 
the measurement of its diameter. Let y denote the volume, x the diameter, 
then> by a well-known mensuration formula, 

y = ^- 

It is required to find the effect of a small error in the measurement of the 



.;. PROBABILITY AND THE THEORY OF ERRORS. 457 

diameter on the calculated volume. Suppose an error dx is committed in the 
measurement, then 

.'/ + dy = kw (x + (1 

= frr{.i"' + H.r-W.1- + :-J.r(</.i-)'- -f ('/.!)';. 

By hypothesis, dx is a very small fraction, therefore, by neglecting the higher 
powers of dx and dividing the result by the original expression 
.// -f dy _ 1 (3? + 3x*dx\ dy dx 
y -5*\ *- J ; y = 8 T- 

Or, the error in the calculated result is three times that made in the 
measurement. Hence the necessity for extreme precautions in measuring 
the diameter. Sometimes, we shall find, it is not always necessary to be so 
careful. 

The same result could have been more easily obtained by the use of 
Taylor's theorem as described above. Differentiate the original expression 
and divide the result by the original expression. We thus get the relative 
error without trouble. 

(2) Criticise the method for the determination of the atomic weight of 
lead from the ratio Pb : O in lead monoxide. 

Let y denote the atomic weight of lead, a the atomic weight of oxygen 
(known). It is found experimentally that x parts of lead combine with one 
part of oxygen, the required atomic weight of lead is determined from the 
simple proportion 

y : a = x : 1 ; or, y = ax ; or, dy = adx ; 

.'.dyly = dxlx (2) 

Thus an error of 1 / in the determination of x introduces an equal error 
in the calculated value of y. Other things being equal, this method of 
finding the atomic weight of lead is, therefore, very likely to give good 
results. 

(3) Show that the result of determining the atomic weight of barium by 
precipitation of the chloride with silver nitrate is less influenced by experi- 
mental errors than the determination of the atomic weight of sodium in the 
same way. 

Assume that one part of silver nitrate requires x parts of sodium (or 
barium) chloride for precipitation as silver chloride. Let a and b be the 
known atomic weights of silver and chlorine. Then, if y denotes the atomic 
weight of sodium, 

y + b : a = .r : 1 ; or, y = ax - 6, 

a = (y + tyl*. 
Differentiating (3) and substituting for y = 23, b = 35-5, 

dy a d? y + l 2-54- 

y = ax - b dx y ' x~ * x ' 

or an error of 1/ in the determination' of chlorine in sodium will introduce 
an error of 2-5% in the atomic weight of sodium. Hence it is a disadvantage 
to have b greater than y. For barium the error introduced is l-5/ instead 

<rf2-5/ . 

(4) If the atomic weight of barium y is determined by precipitation of 
barium sulphate from barium chloride solutions, and a denotes the known 
atomic weight of chlorine, b the known " atomic " weight of .SO 4 , then 



458 HIGHER MATHEMATICS. ; is:;. 

* 

when x parts of barium chloride are converted into one part of barium 
sulphate, 

dy (a - b)dx 

y + a:y+b = l:x;j = (1 _ x} (ax _ b y 

What does this mean ? 

(5) An approximation formula used in the determination of the viscosity 
of liquids is 

T, = Trptr+ISvl, 

where v denotes the volume of liquid flowing from a capillary tube of radius r 
and length I in the time t ; p is the actual pressure exerted by the column of 
liquid. Show that the proportional error in the calculation of the viscosity r? 
is four times the error made in measuring the radius of the tube. 

(6) In a tangent galvanometer, the tangent of the angle of deflection of 
the needle is proportional to the current. Prove that the proportional error 
in the calculated value of the current due to a given error in the reading is 
least when the deflection is 45. 

The strength of the current is proportional to the tangent of the displaced 
angle x, or 

n * /(#) = c tan x ; 

C . dx dy dx 

.'. dy 9 ; or, = 

cos 2 a: ' y sin x . cos x 

To determine the minimum, put 

d fdy\ sin 2 ^ - cos' 2 a? _ ~ . 
dx\ y ) sin 2 .c . cos 2 ic 
.-. sin 2 ic = cos' 2 x, or, sin a; = cosx. 

This is true only in the neighbourhood of 45,* and, therefore, in this region. 
an error of observation will have the least influence on the final result. In 
other words, the best results are obtained with a tangent galvanometer when 
the needle is deflected about 45. 

What will be the effect of an error of O25 in reading a deflection of 42,. 
on the calculated current ? Note that x in the above formula is expressed in 
circular or radian measure (page 494). Hence, 

O25(degrees) = * x ' 2o = -00436(radians). 



y sin x . cos x sin 2x sin 84 
since, from a Table of Natural Sines, sin 84 = -9945. 

(7) Show that the proportional error involved in the measurement of an: 
electrical resistance on a Wheatstone's bridge is least near the middle of the 
bridge. 

Let R denote the resistance, I the length of the bridge, x the distance of 
the telephone from one end. 

.-. y = Rxl(l + x). 

Proceed as above and show that when x = \l (the middle of the bridge), the 
proportional error is a minimum. 

* Table XIV., page 497, sin 45 = cos 45. 



.$ is:;. PROBABILITY AND Till! TIIKoKY OK K 

(8) By AV/r/o/i\s l<iir of nttmctittn, the force of gravitation (g) between 
two bodies varies directly as their respective masses (TO,, w a ) and inversely as 
the square of their distance apart (r). The mass of each body is supposed to 
be collected at its centroid (centre of gravity). The weight of one gram at 
Paris is equivalent to 980-868 dynes. The dyne is the 'unit of force. Hence 
Newton's law. g = nin^njr- (dynes), may be written w = a/r 2 (grams), where 
a is a constant equivalent to /* x H^ x w. 2 x 980-868. Hence show that for 
small changes in altitude dwjw = - 2dr/r. Interpret this result. 

Marek was able to detect a difference of '1 in 500,000,000 when comparing 
the kilogram standards of the Bureau International des Poids et Mesures. 
Hence show that it is possible to detect a difference in the weight of a sub- 
stance when one scale pan of the balance is raised one centimetre higher than 
the other. (Radius of the earth = 6,371,300 metres.) 

(9) In his well-known work on the gravimetric composition of water, Dumas 
determined the weight of hydrogen from the difference in the weight of oxygen 
required to burn up the hydrogen and the weight of water formed. Hence 
verify Dumas' remarks : " ainsi, une erreur de ^^ sur le poids de 1'eau, ou de 
^for sur le poids de 1'oxygene, affecte d'une quantite egale a ^V ou a ^ le poids 
de 1'hydrogene. Que ces erreurs etant dans le meme sens viennent a s'ajouter, 
et 1'on aura des erreurs qui iront a ^ " (Ann. de Chim. et de Phys. [3], 8, 198, 
1843). 

Proportional errors of composite measurements. Whenever a 
result has to be determined indirectly by combining several different 
species of measurements weight, temperature, volume, electro- 
motive force, etc. the effect of a percentage error of, say, 1 / in 
the reading of the thermometer will be quite different from the 
effect of an error of 1 / in the reading of a voltmeter. 

It is obvious that some observations must be made with 
greater care than others in order that the influence of each kind 
of measurement on the final result may be the same. 

If a large error is compounded with a small error, the total 
error is not appreciably affected by the smaller. Hence Ostwald 
recommends, "a variable error may be neglected if it is less than 
one-tenth of the larger, often indeed if it is but one-fifth ". 

EXAMPLES. (1) Joule's relation between the strength of a current C 
(amperes) and the quantity of heat Q (calories) generated in an electric con- 
ductor of resistance R (ohms) in the time / (seconds), is, 

Q = 0-24C-/.'/. 

Show that R and t must be measured with half the precision of C in order to 
have the same influence on Q. 

(2) What will be the fractional error in Q corresponding to a fractional 
error of O'l / in R ? Ansr. 0-001, or 0-1 / . 

(3) What will be the percentage error in C corresponding to 0-02% in Q? 
Ansr. 0-01 / . 



460 HIGHER MATHEMATICS. 183. 

(4) If the density s of a substance be determined from its weights (w, w^ 
in air and water, show that 

ds w l /dw l dw\ 

s ~ w - w\ w l ~ w )' 
Note s = w l l(u- - w t ). 

(5) The specific heat of a substance determined by the method of mixtures 
is given by the formula 



m(6 - 2 ) ' 

where ra is the weight of the substance before the experiment ; 7w a the weight 
of the water in the calorimeter ; c the mean specific heat of water between 
2 and X ; is the temperature of the body before immersion ; X the maximum 
temperature reached by the water in the calorimeter ; 2 the temperature of 
the system after equalisation of the temperature has taken place. Supposing 
the water equivalent of the apparatus is included in m x , what will the effect 
of a small error in the determination of the different temperatures have on the 
result ? 

First, error in X . Show that 

ds/s = - d0j/(0 2 - ej. 

If an error of say 0'1 is made in a reading and 2 - O l = 10, the error in the 
resulting specific heat is about 1 / . If a maximum error of O'l / is to be per- 
mitted, the temperature must be read to the 0*01. 

Second, error in 0. Show that 

ds/s = dOl(6 - 2 ). 

If a maximum error in the determination of s is to be O'l / , when 6- 2 = 50, 
6 must be read to the 0'04. If an error of 0'1 is made in reading the tem- 
perature and e - 6 2 50, show that the resulting error in the specific heat 
will be 0-2/ . 

Third, error in 2 . Show that 

ds/s = d0 2 /(0 2 - ej + de 2 /(0 - 2 ). 

If the maximum error allowed is Ol / and 2 - e l = 10, - 0j = 50, show 
that 2 must be read to the T F ; while if an error of 0-1 is made in the 
reading of 2 , show that the resulting error in the specific heat is 0'5 / . 

(6) In the preceding experiment, if w : = 100 grams, show that the 
weighing need not be taken to more than the 0*1 gram for the error in s to be 
within O'l / ; and for m, need not be closer than 0-5 gram when m is about 
50 grams. 

Since the actual errors are proportional to the probable errors, 
the most probable or mean value of the total error dn, is obtained 
from the expression 



from (16), $ 179, page 444. Note the squared terms are all positive. 
Since the errors are fortuitous, there will be as many positive as 
negative paired terms. These will, in the long run, approximately 
neutralize each other. Hence (3). 



PROBABILITY AND THE THEORY OF KKKoRS. 461 

EXAMPLES. (1) Divide equation (8) by n", it is then easy to show that 

(dQIQf = (2dC/C) 2 + (dRIRy + (<////), 

from the preceding set of examples. Hence show that the fractional error in 
Q, corresponding to the fractional errors of 0*03 in C, - 02 in H and 0'03 in t, 
is 0-07. 

(2) The regular formula for the determination of molecular weight of a 
substance by the freezing point method, is M = Kw/0, where A" is a constant, 
.17 the required molecular weight, -w the weight of the substance dissolved in 
100 grams of the solvent, 6 the lowering of the freezing point. In an actual 
determination, w -5139, 6 = -295, K = 19 (Perkin and Kipping's Organic 
Clu'mistry), what would be the effect on M of an error of '01 in the deter- 
mination of w, and of an error of '01 in the determination of 6 ? 

Also show that an error of -01 in the determination of affects M to an 
extent of - 3*25, while an error of '01 in the determination of w only affects 
^[ to the extent of -19. Hence show that it is not necessary to weigh to more 
than O'Ol of a gram. An illustration of the need of " scientific perspective '* 
in measuring the different components of a composite result. 

From (16), 179, page 444, when the effect of each observation 
on the final result is the same, the partial differential coefficients are 
all equal. If u denotes the sum of n observations, a lt a. 2 , . . ., a n , 

u = a x + a 2 + . . . + a n , 
^u ^u 
daj ^a 2 

But in order that the actual errors affecting each observation may 
be the same, 

cZj = da 2 = . . . = da n = du/ Jn ; . . (4) 

da, da* da n du 1 

from (3), or, -' - ^ - . . . - - ---^ . (5) 

EXAMPLES. (1) Suppose the greatest allowable fractional error in Q 
(preceding examples) is 0-5 / , what is the greatest percentage error in each 
of the variables C, R, t, allowable under equal effects ? Here, 

2dC/C = dR/R = dtjt = -006/ s ; 3. 
Ansr. 0-22 for R and t, '11 / for C. 

(2) If a volume v of a given liquid flows from a long capillary tube of 
radius r and length I in t seconds, the viscosity of the liquid is TJ = vpr*t/8vl t 
where p denotes the excess of the pressure at the outlet of the tube over 
atmospheric pressure. What would be the errors dr, di\ dl, dt, dp, necessary 
under equal effects to give rj with a precision of -1 / ? Here, 

dplp = dt/t = 4dr/r = - dv/v = - dl/l = -001/ v'S = -00045. 
It is now necessary to know the numerical values of p, t, v, r, I, before 
dp, dt, . . . can be determined. Thus, if r is about 2 mm., the radius must 
be measured to the -0022 mm. for an error of -1 / in TJ. 

It has been shown how the best working conditions may be determined by 



462 HIGHER MATHEMATICS. g 184. 

a study of the formula to which the experimental results are to be referred. 
The following is a more complex example. 

(3) The resistance i of a cell is to be measured. Let C lf C 2 respectively 
denote the currents produced by the cell when working through two known 
external resistances i\ and ? 2 , and let R v R 2 , be the total resistances of the 
circuit, E the electromotive force of the cell is constant. It is known (see 
your textbook : Practical Physics), 

i = (C 2 r 2 - C^KC^ - C 2 ) (6) 

What ratio R l : R 2 will furnish the best result ? From Ohm's law, E = CR, 
E being constant, C l :C 2 = R 2 : R r As usual, (4) above 



Find values for 'dil'dC 1 and di/9C 2 from (7), and put R l for i\, R.-, for r. 2 . Thus, 

Substitute this result in (7). 

1. If a mirror galvanometer is used, dC l = dC 2 = dC (say) constant. 

... (di) z = (R^R^ - RfRJ) (dC) 2 IE*(Rz - RJ*. 
Substitute x = R 2 : R lt 

For a minimum error 
d 



.-. x = 2-2 approx. ; or, R 2 - 2-2^ ; or, Cj = 2-2C 2 . Substitute these values 
in (6), 

di = v20 . R* . dCJE, 

which shows that the external resistance R v should be as small as is consistent 
with the polarisation of the battery. 

2. If a tangent galvanometer is used, dC/C is constant. Hence substitute 
C-i - ER l and C 2 = ER 2 in (8), 



From this it can be shown there is no best ratio R : R r If the last expression 
is written 



it follows that the error di increases as R 1 increases, and as R., diminishes. 
Hence R 2 should be made as large and R l as small as is consistent with the 
range of the galvanometer and the polarisation of the battery. 



184. Observations of Different Degrees of Accuracy. 

Hitherto it has been assumed that the individual observations 
of any particular series, are equally reliable, or that there is no 
reason why one observation should be preferred more than an- 
other. As a general rule, measurements made by different 
methods, by different observers, or even by the same observer at 






< IS4. PROBABILITY AND TIIK TIIKulIY OF KUKoi;- 

dil'lerent times,* are not liable to the same errors. Some results 
11. more trustworthy than others. In order to fix this idea, 
suppose that twelve determinations of the capacity of a flask by 
tlif same method, gave the following results: six measurements 
each 1'6 litres; four, 1-4 litres; and two, 1/2 litres. The numbers 
6, 4, 2, represent the relative values of the three results 1-6, 1-4, 
1-2, because the measurement 1*6 has cost three times as much 
labour as 1*2. The former result, therefore, is worth three times 
as much confidence as the latter. In such cases, it is customary 
to say that the relative practical value, or the weight of these three 
sets of observations is as 6:4:2, or, what is the same thing, as 
3:2:1. In this sense, the weight of an observation, or set of 
observations, represents the relative degree of precision of that 
observation in comparison with other observations of the same 
quantity. 'It tells us nothing about the absolute precision (h) of 
the observations. 

It is shown below that the weight of an observation is, in 
theory, inversely as its probable error ; in practice, it is usual to 
assign arbitrary weights to the observations. For instance, if one 
observation is made under favourable conditions, another under 
adverse conditions, it would be absurd to place the two on the 
same footing. Accordingly, the observer pretends that the best 
observations have been made more frequently. That is to say, 
if the observations a lf a. 2 , . . ., a,,, have weights p^ p. 2 , . . ., p n , 
respectively, the observer has assumed that the measurement a l 
has been repeated p l times with the result a lt and that a,, has been 
repeated p tl times with the result a n . 

To take a concrete illustration, Morley has made three accurate 
series of determinations of the density of oxygen gas with the 
following results : 

I. 1-42879 -000034; II. 1 -42887 "000048; III. 1 -4291 7 '000048. 
(" On the densities of oxygen and hydrogen and on the ratio of 



* I am reminded that Dumas, discussing the errors in his great work on the gravi- 
metric composition of water, alluded to a few pages back, adds the remarks : " The 
length )t' time required for these operations compelled me to prolong the work far into 
the night, generally finishing with the weighings about 2 or 3 o'clock in the morniiu. 
This may be the cause of a substantial error, for I dare not venture to assert that >urh 
weighings deserve as much confidence as it they had been performed under more 
favourable conditions and by an observer not so worn out with fatigue, the inevitable 
result of fifteen to twenty hours continued attention." 



464 HIGHER MATHEMATICS. $ 184. 

their atomic weights," Smithsonian Contributions to Knowledge ^ 
(980), p. 55, 1895.) The probable errors of these three means 
would indicate that the first series were worth more than the 
second. For experimental reasons, Morley preferred the last 
series, and gave it double weight. In other words, Morley pre- 
tended that he had made four series of experiments, two of which 
gave 1-42917, one gave 1-42879, and one gave 1-42887. The result 
is that 1-42900, not 1-42894,* is given as the best representative 
value of the density of oxygen gas. 

The product of an observation or of an error with the weight 
of the observation, is called a weighted observation, or a weighted 
error as the case might be. 

The practice of weighting observations is evidently open to 
some abuse. It is so very easy to be influenced rather by the differ- 
ences of the results from one another, than by the intrinsic quality 
of the observation. This is a fatal mistake. 

1. The best value to represent a number of observations of equal 
weight, is their arithmetical mean. 

If P denotes the most probable value of the observed magnitudes a^ <i. 2r 
. . ., n , then P - a lt P - a 2 , . . ., P - a n , represent the several errors in the 
n observations. From the principle of least squares these errors will be a. 
minimum when 

(P - Oj) 2 + (P - 2 ) 2 + + ( p ~ n) 2 = a minimum. 

Hence, page 434, P = (a^ + a 2 + . . . + a n )jn, . . . . (1^ 

or the best representative value of a given series of measurements of an un- 
known quantity, is the arithmetical mean of the n observations, provided that 
the measurements have the same degree of confidence. 

2. The best value to represent a number of observations of 
different weight, is obtained by multiplying each observation by its 
weight and dividing the sum of these products by the sum of their 
different weights. 

With the same notation as before, let p^ p*, . . ., p n , be the respective 
weights of the observations a lt 2 , . . ., a n . From the definition of weight, 
the quantity a x may be considered as the mean of p l observations of unit 
weight ; 2 , the mean of p% observations of unit weight, etc. The observed 
quantities may, therefore, be resolved into a series of fictitious observations- 
all of equal weight. Applying the preceding rule to each of the resolved 
observations, the total number of standard observations of unit weight will 

* See formula (5). 



isi. PIJnllAIUUTY AND THK THKnliY <>! KIM 

bo /, + />+...+ )> ; the sum of the ;>, standard observations of unit weight 
v.ill be /v/, ; the sum of />. 2 standard observations, p.fa, etc. Hence, from (1), 
the most probable value of a series of observations of different, weights is 

] = P^ a ^ + /*a + . . . + p n a n / 2 \ 

Pi + Pt + + Pn 

Note the formal resemblance between this formula and that for finding the 
centre of gravity of a system of particles of different weights arranged in a 
straight line. 

Weighted observations are, therefore, fictitious results treated 
as if they were real measurements of equal weight. With this 
convention, the value of P' in (2) is an arithmetical mean some- 
times called the general or probable mean. 

3. The weight of an observation is inversely as the square of its 
probable error. 

Let a be a set of observations whose probable error is R and whose weight 
is unity. Let 2hi P& Pn an< i r u r 2 > ? ' ^ e ^e respective weights and 
probable errors of a series of observations rt lf 2 , . . ., a,,, of the same quantity. 
By definition of weight, a x is equivalent to i\ observations of equal weight. 
From (16), page 444, 

r, = El v^ ; or, Pl 



EXAMPLES. (1) If n observations have weights p p& ., p nt show that 



Differentiate (2) successively with respect to j, ,,,... and substitute the 
results in (16), page 444. 

(2) Show that the mean error of a series of observations of weights p^ p. 2t 
,>.., is 



Hint. Proceed as in 178 but use px 2 and pv z in place of .t 2 and y 2 respectively, 
If the sum of the weights of a series of observations is 2(/>) = 40, and the sum 
of the products of the weights of each observation with the square of its 
deviation from the mean of nine observations is 2(px 2 ) = '3998, show that 
M = 0-035. 

(3) The probable errors of four series of observations are respectively 1-2, 
0-8, 0-9, 1-1, what are the relative weights of the corresponding observations? 
Ansr. 7 : 16 : 11 : 8. Use (3). 

(4) Determinations of the percentage amount of copper in a sample of 
malachite were made by a number of chemical students, with the following 
results : (1) 39-1 ; (2) 38-8, 38'7, 38-6 ; (3) 39-9, 39-1, 39-3 ; (4) 37'7, 37'9. If 
these analyses had an equal degree of confidence, the mean, 38-8, would best 
represent the percentage amount of copper in the ore formula (1). But the 
analyses are not of equal value. The first was made by the teacher. To this 
we may assign an arbitrary weight 10. Sets (2) and (3) were made by two 

GG 



466 HIGHER MATHEMATICS. $ 184. 

different students using the electrolytic process. Student (2) was more ex- 
perienced than student (3), in consequence, we are led to assign to the former 
an arbitrary weight 6, to the latter, 4. Set (4) was made by a student pre- 
cipitating the copper as CuS, roasting and weighing as CuO. The danger 
of loss of CuS by oxidation to CuSO during washing, leads us to assign to 
this set of results an arbitrary weight 2. From these assumptions, show that 
38-94 best represents the percentage amount of copper in the ore. For the 
sake of brevity use values above 37 in the calculation. From formula (2), 
108-8/56 = 1-94. Add 37 for the general mean. 

It is unfortunate when so fantastic a method has to be used for calculating 
the most probable value of a " constant of Nature," because a redetermination 
is then urgently required. 

(5) Rowland (Proc. Amer. Acad., 15, 75, 1879) has made an exhaustive 
study of Joule's determinations of the mechanical equivalent of heat, and 
he believes that Joule's several values have the weights here appended in 
brackets: 442-8 (0); 427'5 (2); 426-8 (10); 428-7 (2); 429-1 (1); 428-0 (1) ; 
425-8 (2) ; 428-0 (3) ; 427-1 (3) ; 426-0 (5) ; 422-7 (1) ; 426-3 (1). Hence Rowland 
concludes that 426-9 best represents the result of Joule's work. Verify this. 
Notice that Rowland rejects the number 442-8 by giving it zero weight. 

4. To combine several arithmetical means each of which is affected 
.with a known probable (or mean) error, into one general mean. 

One hundred parts of silver are equivalent to 

49*5365 -013 of NH^Cl, according to Pelouze ; 
49-523 -0055 Marignac ; 

49-5973 -0005 Stas (1867) ; 

49-5992 + '00039 Stas (1882), 

^where the first number represents the arithmetical mean of a series 
of experiments, the second number the corresponding probable 
error. How are we to find the best representative value of this 
series of observations? The first thing is to decide what weight 
shall be assigned to each result. Individual judgment on the 
"internal evidence" of the published details of the experiments 
is not always to be trusted. Nor is it fair to assign the greatest 
weight to the last two values simply because they are by Stas. 

Meyer and Seubert, in a paper Die Atomgewichte der Elements, 
aus der Originalzahlen neu berechnet (Leipzig, 1883), weighted 
each result according to the mass of material employed in the 
determination. They assumed that the magnitude of the errors 
of observation were inversely as the quantity of material treated. 
That is to say, an experiment made on 20 grams of material is 
supposed to be worth twice as much as one made on 10 grams. 
This seems to be a somewhat gratuitous assumption. 






.* IS4. PROBABILITY AND T1IK THKuKY <>! KUKOltS. 

One way of treating this delicate question is to assign to each 
arithmetical mean a weight inversely as the square of its mean 
error. Clark in his " Recalculation of the Atomic Weights " 
(Smithsonian Miscellaneous Collections (1075), 1897) employed 
the probable error. Although this method of weighting did not 
suit Morley in the special case mentioned on page 463, Clark 
considers it a safe, though not infallible guide. 

Let A, B, C, . . ., be the arithmetical mean of each series of 
experiments ; a, b, c, . . ., the respective probable (or mean) errors, 
then, from (2), 

A_ B C_ 
a 2 6 2 c 2 



General Mean 



Probable Error = + 



(5) 



EXAMPLES. (1) From the experimental results just quoted, show that 
the best value for the ratio 

Ag : NH 4 Cl is 100 : 49-5983 -00031. 

Hint. Substitute A = 49-5365, a = -013 ; B = 49 '523, b = -0055 ; C = 49-5973, 
c = -0005 ; D = 49-5992, d = -00039, in equations (5). 

(2) The following numbers represent the most trustworthy results yet pub- 
lished for the atomic weight of gold (H = 1) : 195-605 -0099 ; 195-711 '0224 ; 
195-808 -0126; 195-624 -0224; 195-896 '0131 ; 195-770 -0082. Hence 
show that the best representative value for this constant is 196'743 -0049. 

(3) In three series of determinations of the vapour pressure of water 
vapour at Regnault found the following numbers : 

I. 4-54; 4-54; 4-52; 4-54; 4-52; 4-54; 4 '52 ; 4'50 ; 4-50; 4-54. 
II. 4-66 ; 4-67 ; 4-64 ; 4-62 ; 4-64 ; 4-66 ; 4-67 ; 4-66 ; 4-66. 
III. 4-54; 4-54; 4-54; 4-58 ; 4-58; 4-57; 4-58. 

Show that the best representative value of series I. is 4-526, with a probable 
error 0-0105 ; series II., 4-653, probable error + 0-0105 ; series III., 4-561, 
probable error 0-0127. The most probable value of the vapour pressure of 
aqueous vapour at is, therefore, 4-582, with an equal chance of its possessing 
an error greater or less than -0064. 

" As a matter of fact the theory of probability is of little or no importance, 
when the * constant ' errors (otherwise known as ' systematic ' errors) are 
greater than the accidental errors. Still further, this use of the probable 
error cannot be justified, even when the different series of experiments are 
only affected with accidental errors, because the probable error only shows how 
UNIFORMLY an experimenter JMS conducted a certain process, and not how 
suitable that process is for tJie required purpose. In combining different se^s 
of determinations it is still more unsatisfactory to calculate the probable 



468 HIGHER MATHEMATICS. 184. 

error of the general mean by weighting the individual errors according to 
Clark's criterion when the probable errors differ very considerably among 
themselves. For example, Clark (I.e., page 126) deduces the general mean 
136-315 + -0085 for the atomic weight of barium from the following results : 

136-271 -0106 ; 136-390 -0141 ; 135-600 -2711 ; 136-563 -0946. 
The individual series here deviate from the general mean more than the 
magnitude of its probable error would lead us to suppose. The constant 
errors, in consequence, must be greater than the probable errors. In such a 
.ease as this, the computed probable error + -0085 has no real meaning, and 
we can only conclude that the atomic weight of barium is, at its best, not 
known more accurately than to five units in the second decimal place." 
(Paraphrased from Ostwald's critique on Clark's work (I.e.) in the Zeitschrift 
fur physikalisclie Chemie, 23, 187, 1897.) 

5. Mean and probable errors of observations of different degrees 
of accuracy. 

In a series of observations of unequal weight the mean and probable errors 
of a single observation of unit weight are respectively 

3k 

- (6) 

The mean of a series of observation of unequal weight has the respective 
mean and probable errors 



EXAMPLE. An angle was measured under different conditions fourteen 
times. The observations all agreed in giving 4 15', but for seconds of arc 
the following values were obtained (the weight of each observation is given in 
brackets) : 45"-00 (5) ; 31 "-'25 (4) ; 42"-50 (5) ; 45"-00 (3) ; 37"-50 (3) ; 38"-33 (3) ; 
27"-50 (8) ; 43"'33 (3) ; 40"-63 (4) ; 36"-25 (2) ; 42"-50 (3) ; 39"-17 (3) ; 45"-00 (2) ; 
40"-83 (3). Show that the mean error of a single observation of unit weight 
is 9"'475, the mean error of the mean 39"-78 is l"-397. Hint. 2(p) = 46 ; 
= 1167-03 ; n = 14 ; 2(pa = 1830-00. 



The mean and probable errors of a single observation of weight p are 



respectively. 

EXAMPLE. In the preceding example show that the mean error of an 
observation of weight (2) is 6"-70 ; of weight (3) is + 5"-47 ; of weight (4) 
4"-74 ; and of weight (5) 4"-24. 

6. The principle of least squares for observations of different 
degrees of precision states that ''the most probable values of the 
observed quantities are those for which the sum of the weighted 
squares of the errors is a minimum," that is, 

p-fv-f + p 2 2 v 2 2 + . . . + p n 2 v n 2 = a minimum. 






^ is:,. PROBABILITY AND THK THKnuv OF KKKOKs HJ'.i 

An error v is the deviation of an observation from the arithmetical 
mean of n observations ; a " weighted square " is the product of 
the weight p and the square of an error v (see 106). 

185. Observations Limited by Conditions. 

On adding up the results of an analysis, the total weight of the 
constituents ought to be equal to the weight of the substance itself ; 
the three angles of a plane triangle, must add up to exactly 180 ; 
the sum of the three angles of a spherical triangle always equal 
180 + the spherical excess ; the sum of the angles of the nor- 
mals on the faces of a crystal in the same plane must equal 360 . 
Measurements subject to restrictions of this nature, are said to 
be conditioned observations. The number of conditions to be 
satisfied is evidently less than the number of observations, other- 
wise the value of the unknown could be deduced from the 
conditions, without having recourse to measurement. 

In practice, measurements do not come up to the required 
standard, the percentage constituents of a substance do not add 
up to 100 ; the angles of a triangle are either greater or less than 
180. Only in the ideal case of perfect accuracy are the conditions 
fulfilled. It is sometimes desirable to find the best representative 
values of a number of imperfect conditioned observations. The 
method to be employed is illustrated in the following examples. 

EXAMPLES. (1) The analysis of a compound gave the following results : 
37-2 / of carbon, 44-1% of hydrogen, 19-4 / of nitrogen. Assuming each 
determination is equally reliable, what is the best representative value of the 
percentage amount of each constituent ? Let C, H, N, respectively denote 
the percentage amounts of carbon, hydrogen, and nitrogen required, then, 

C + tf = 100 - 2VEE 100 - 19-4 = 80-6. 
Hence, 2C + H = 117-8 ; C + 2H = 124-7. 

Solve the last two simultaneous equations in the usual way. Ansr. C = 36-97 / ; 
H= 43-86 / ; N= 19-17 / . Note that this result is quite independent of 
any hypothesis as to the structure of matter. The chemical student will 
know a better way of correcting the analysis. This example will remind us 
how the atomic hypothesis introduces order into apparent chaos. Some 
analytical chemists before publishing their results, multiply or divide their 
percentage results to get them to add up to 100. In some cases, one consti- 
tuent is left undetermined and then calculated by difference. Both practices 
are objectionable in exact work. 

(2) The three angles of a triangle A, B, C, were measured with the result 
that .4 = 51 ; B = 94 20' ; C = 34 56'. Show that the most probable values 
of the unknown angles are A = 51 56' ; B = 94 12' ; C = 34 52'. 



470 HIGHER MATHEMATICS. 185. 

(3) The angles between the normals on the faces of a cubic crystal were 
found to be respectively a = 91 13' ; = 89 47' ; y = 91 15' ; 5 = 89 42'. 
What numbers best represent the values of the four angles ? Ansr. a = 90 
43' 45" ; = 89 17' 45" ; 7 = 90 0' 45" ; 5 = 89 57' 45". 

(4) The three angles of a triangle furnish the respective observation 
equations : 

A = 36 25' 47" ; B = 90 36' 28" ; C = 52 57' 57" ; 
the equation of condition requires that 

A + B + C - 180 = (1) 

Let #!, a? 2 , x s , respectively denote the errors affecting A, B, C, then we must 
have 

x l + x 2 + x s = - 12 (2) 

i. If the observations arc equally trustworthy, 

Xi = x 2 = x 3 = k, (3) 

say. Substitute this value of x lt x 2 , x st in (2), and we get 

3k + 12 = ; or, k = - 4 ; 

.-. A = 36 25' 43" ; B = 90 36' 24" ; C = 53 57' 53". 
The formula for the mean error of each observation is 



J. *<*> 

Mn - w + 






where w denotes the number of unknown quantities involved in the n ob- 
servation equations ; q denotes the number of equations of condition to be 
satisfied. Consequently the w unknown quantities reduce to w - q inde- 
pendent quantities. 2(v 2 ) denotes the sum of the squares of the differences 
between the observed and calculated values of A, B, C. Hence, the mean 
error = + \/48 = 6" -93. 

ii. If tlie observations have different weights. Let the respective weights 
of A, B, C, be p l = 4 ; p z = 2 ; p s = 3. It is customary to assume that the 
magnitude of the error affecting each observation will be inversely as its 
weight. (Perhaps the reader can demonstrate this principle for himself.) 
Instead of (3), therefore, we write 

x l = k; x. 2 = k x s = $k ..... (5) 
From (2), therefore, 

13fc + 144 = ; k = - 11-07 ; x 1 = - 2"-77 ; x >2 = - 5"-54 ; x : , = - 3"'69. 



, = Mean error = __, ... (6) 
or in = 11-52. 

The mean errors m^, ; 2 , w 3 , respectively affecting a, 6, c, are 

m in in 

ni.=+ , ; iito = +-._; m, = + T-. . . (7) 

- x^' - x y " - j p 

Hence 

.4=36 25' 44"-235"-76; 5 = 90 36' 22"-468"-15 ; C = 52 57' 53"'316"'65. 

It is, of course, only permissible to reduce experimental data in 
this manner when the measurements have to be used as the basis 
for subsequent calculations. In every case the actual measure- 
ments must be stated along with the " cooked " results. 



$ ISO. l'R<>P>AinUTY AM) TIIK TIIKniiV OF BBBOBE 17! 

186. Gauss' Method of Solving a Set of Linear Observation 

Equations. 

In continuation of 106, let x, y, z, represent the unknowns to be 
evaluated, and let a lt a. 2 , . . .,b l ,b. 2 , . . .,c lt c. 2 , . . ., /i',, //,, . . ., 
represent actual numbers whose values have been determined by 
the series of observations set forth in the following observation 
equations : 



a> 2 x + b 2 y + c, 2 z = R 2 
a.jK + b z y + c z z = \R 3 ; I 
ax + by + cz = R 4 . J 

If only three equations had been given, we could easily calculate 
the corresponding values of x, y, z, by the method of 165, but 
these values would not necessarily satisfy the fourth equation. 
The problem here presented is to find the best possible values of 
x, y, z, which will satisfy the four given observation equations. 
We have selected four equations and three unknowns for the sake 
of simplicity and convenience. Any number may be included in 
the calculation. But sets involving more than three unknowns are 
comparatively rare. 

We also assume that the observation equations have the same 
degree of accuracy. If not, multiply each equation by the square 
root of its weight, as in example (3) below. This converts the 
equations into a set having the same degree of accuracy. 

First. To convert the observation equations into a set of normal 
equations solvable by ordinary algebraic processes. 

Multiply the first equation by a lt the second by a.,, the third by 
a 3 , and the fourth by 4 . Add the four results. Treat the four 
equations in the same way with b lt b. 2 , b. 3 , 6 4 , and with c lf c 2 , c 3 , c 4 . 
Now write, for the sake of brevity, 
[aa\ = V + a* + o s 2 + a 4 2 ; [bb], = b,- + b^ + V + b* ; 



aj = a l jj + a 22 + a 3 ^ + a 4 

and likewise for [cc] v [bc]^ [cR] r The resulting equations are 
[aa\x + [ab\y + [ac\z = [aR\ ; j 
[db\x + [bb] iy + \bc\z = [bB^ ; - . . (2) 
[c]> + [be]# + [cc]> = [cR] r \ 

These three equations are called normal equations (first set) in 

x, y, z. 



472 HIGHER MATHEMATICS. 186. 

Second. To solve the normal equations. We can determine 
the values of x, y, z, from this set of simultaneous equations (2) by 
any method we please, determinants ( 165), cross -multiplication, 
indeterminate multipliers, or by the method of substitution.* The 
last method is adopted here. 

Solve the first normal equation for x, thus, 
[oH \ac^ [oB], 

- - h 



Substitute this value of x in the other two equations for a second 
set of normal equations in which the term containing x has dis- 
appeared. 



For the sake of simplicity, write 
[bb], = 



, = [WZ], - 
The second set of normal equations may now be written : 



[bc^y + [cc] 2 = [cB] 2 . 
Solve the first of these equations for y, 



Substitute this in the second of equations (4), and we get a third 
set of normal equations, 



which may be abbreviated into 

[cc],z = [cR],. . (6) 

Hence, z = [^ ..... (7) 

OL 

, [6c] 2 , . . ., [cc] 3 , . . . are called auxiliaries. 



* The equations cannot be solved if any two are identical, or can be made identical 
by nrmltiplying through with a constant. 



i LSI;. l'l;ii|;\l',II.ITY AND IIIK I IIKuKV i)F KIMH >|;-. 17:1 

Equations (3), (5), (7), collectively constitute a set of elimination 
equations : 






1 [<*],' 



z = 

~ 



The last equation gives the value of z directly ; the second gives 
the value of y when z is known, and the first equation gives the 
value of x when the values of y and 2 are known. 

Note the symmetry of the coefficients in the three sets of normal 
equations. Hence it is only necessary to compute the coefficients 
of the first equation in full. The coefficients of the first horizontal 
row and vertical column are identical. So also the second row and 
second column, etc. 

The formation and solution of the auxiliary equations is more 
tedious than difficult. Several schemes have been devised to lessen 
the labour of calculation as well as for testing the accuracy of the 
work. These we pass by. 

Third. The weights of the values of x, y, z. Without entering 
into any theoretical discussion, the respective weights of z, y, and x, 
are given by the expressions : 



\bb\j [calJftftlo 

p. - M. ; ,. - P.LJI ; ,, . tfaffSfa&R . 

Fourth. The mean errors affecting the values of x, y, z. Let 
ax + b + cz - E = v ; 



Let M denote the mean error of any observed quantity of unit 
weight, 



M=J-2-L for equal weight ; 

~^n-w v t (1Q) 

.V = + / 2 (F 1 ''-) for unequal weights, I 

" \ n - iv 

where n denotes the number of observation equations, w the number 
of quantities x, y, z, . . . Here, w = 3, n = 4. Let M x , M^ M t , 
respectively denote the mean errors respectively affecting x, y, z. 



474 



HIGHER MATHEMATICS. 



8 186. 



EXAMPLES. (1) Find the values of the constants a and b in the formula 

y = a + bx, (12). 

from the following determinations of corresponding values of x and y : 

y = 3-5, 5-7, 8-2 10-3, . . .; 

when x = 0, 88, 182, 274, . . . ; 

We want to find the best numerical values of a and b in equations (12). Write 
x for a, and y for 6, so as to keep the calculation in line with the preceding 
discussion. The first set of normal equations is obviously 

[aa\x + \ab\y = \aR\ ; and [ab\x + \bb\y = \bE\. 

But ..-HSkr + Baj-...^!^ 

loaf* [oak' [66V 

Again, \aa\ = 4 ; \bb\ = 115,944 ; [ab\ = 544 ; [aR\ = 27*7 ; [bR\ = 4,816-2 ; 
[bb\ = 4,853-67 ; \bR\ = 115,951-4. 

x = 3-52475 ; ij = 0-02500 ; 

or, reconverting x into a, and y into b, (12) is to be written, 
y = 3-525 + 0-025?. 



b. 
(t 




Difference between 
Calculated and 


Square of Difference 
between Calculated 
and Observed. 


Calculated. 


Observed. 


Observed. 


3 '525 
88 5-725 
182 8-075 
274 10-375 

1 


3-5 
5-7 
8-2 
10-3 


+ 0-025 
+ 0-025 
- 0-125 
+ 0-075 


0-000625 
0-000625 
0-015625 
0-005625 




0-0225 



M = + 0-106. 
41,960 ; M b = -106/ 



Weight of b = p y = [bb] 2 = 41,960 ; M b = -106/ V41,960 = -0004. 
Weight of a = Px = E||i m i-5; j/ /t = + . 106 / v'1^5 = -087. 



= 21 ;\ 
= 14. J 



(2) The following equations were proposed by Gauss to illustrate the above 
method [Gauss' Tlieoria motus corporum co.clestium (Hamburg, 1809) ; Gauss' 
Werke, 7, 240, 1871] : 

x - y + 2z = 3 ; 4.u + y + 
3x + 2y - 5z = 5 ; - .c + 3y + 
Hence show that x = + 2-470 ; y = + 3-551 ; z = + 1-916 . 2(v 2 ) = 0804 ; 
AT = 284 ; p x = 246 ; p y = 136 ; p z = 539 ; M x = -057 ; M y = -077 ; 
M z = -039. Hint. The first set of normal equations is 

27 + 6y = 88; 6x + I5y + z = 10: y + 5z = 107: 

(3) The following equations were proposed by Gauss (I.e.) to illustrate his 
method of solution : 

x y + 2,2 3, with weight 1 
3x + 2y - 5z - 5, 1 

4z + y + 4z = 21, 1 

- 2,r + 6y + 62 = 28, 



(14) 



$ 1*7. I'ROnAlHUTY AND TIIK TIIKnUY OF BRROR& IT-". 

By the rule, multiply the last equation by %/f = and we get set (13). Show 
that x = + 2-47 with a weight 24-6 ; y = + 3-55 with a weight 13-6 ; and 
z = + 1-9 with a weight 53-9. It only remains to substitute these values of 
x, y, z, in (14) to find the residuals r. Hence show that M = + 295. Proceed 
as before for Af x , M y , M t . 

(4) The length (1) of a seconds pendulum at any latitude />, is given by 
Clairaut's equation : 

I = L + A sin 2 !/, 

where L n and A are constants to be evaluated from the following observa- 
tions : 

L = 0', 18 27', 48 24', 58 15', 67 4' ; 
I = 0-990564, 0-991150, 0-993867, 0-994589, 0*995325. 
Hence show that 

I = 0-990555 + 0-005679 sin 2 . 
Hint. The normal equations are, 

x + 0-44765 y = 0-993099 ; x + 0-70306 y = 0-994548. 

The above is based on the principle of least squares. A quicker method, 
not so exact, but accurate enough for most practical purposes, is due to Mayer. 
We can illustrate Mayer's method by means of equations (13). 

First make all the coefficients of x positive, and add the results to form a 
new equation in x. Similarly for equations in ij and z. We thus obtain, 
9x - y - iz = 15 ; j 
5x + 7y =37;}- (15) 

.,; + y + U Z = 33. ) 

Solve this set of simultaneous equations by algebraic methods and we get 
x = 2-485; y = 3*511 ; z = 1-929. Compare these values of x, y, z, with the 
best possible values for these magnitudes obtained in example (2). 



187. When to Reject Suspected Observations. 

There can be no question about the rejection of observations 
which include some mistake, such as a wrong reading of the 
eudiometer or burette, a mistake in adding up the weights or a 
blunder in the arithmetical work, provided the mistake can be 
detected by check observations or calculations. Sometimes a 
most exhaustive search will fail to reveal any reason why some 
results diverge in an unusual and unexpected manner from the 
others. It has long been a vexed question how to deal with 
abnormal errors in a set of observations, for these can only lu> 
conscientiously rejected when the mistake is perfectly obvious. 
It would be a dangerous thing to permit an inexperienced or 
biassed worker to exclude some of his observations simply because 
they do not fit in with the majority. " Above all things," said 



476 HIGHER MATHEMATICS. $ 187. 

the late Prof. Holman in his Discussion on the Precision of Mea- 
surements (Wiley & Sons, New York, 1901), "the integrity of the 
observer must be beyond question if he would have his results 
carry any weight, and it is in the matter of the rejection of doubt- 
ful or discordant observations that his integrity in scientific or 
technical work meets its first test. It is of hardly less importance 
that he should be as far as possible free from bias due either to 
preconceived opinions or to unconscious efforts to obtain concordant 
results." 

Several criteria have been suggested to guide the investigator 
in deciding whether doubtful observations shall be included in the 
mean. Such criteria have been deduced by Chauvenet, Hagen, 
Stone, Pierce, etc. None of these tests, however, is altogether 
satisfactory. Chauvenet's criterion is perhaps the simplest to 
understand and most convenient to use. It is an attempt to 
show, from the theory of probability, that reliable observations 
will not deviate from the arithmetical mean beyond certain 
limits. 

From (2) and (6), 178, 

r = 0-4769//i = 0-6745 \^(v 2 )f(n - 1). 

If x = rt, where rt represents the number of errors less than x which may be 
expected to occur in an extended series of observations when the total number 
of observations is taken as unity, r represents the probable error of a single 
observation. An}- measurement containing an error greater than ,r is to be 
rejected. If n denotes the number of observations and also the number of 
errors, then nP indicates the number of errors less than rt, and n(l - P) the 
number of errors greater than the limit rt. If this number is less than ^, any 
error rt will have a greater probability against than for it, and, therefore, 
may be rejected. 

The criterion for the rejection of a doubtful observation is, therefore, 
xfr = t ; = ra(l - P) ; 

271-1 2 it 
whence P = =r -=r I e~ * dt (1) 

\irJ o 

By a successive application of these formulae, two or more doubtful results 
may be tested. 

The value of t, or, what is the same thing, of P, and hence also of n, can 
be read off from the table of integrals, page 515 (Table XI.). Table XII. con- 
tains the numerical value of xfr corresponding to different values of n. 

EXAMPLES. (1) The result of 13 determinations of the atomic weight of 
oxygen made by the same observer is shown in the first column of the sub- 
joined table. Should 19-81 be rejected ? Calculate the other two columns of 
the table in the usual way. 



187. PROBABILITY AND THK IHI-.oKY OP ERRORS 177 



Observation. 


X. 


*a. 


vntion. 


X. 


&. 


15-96 


- 0-26 


0-0676 


15-88 


- 0-34 


0-1166 


19-81 


+ 3-59 


12-8881 


15-86 


- 0-36 


0-1296 


15-95 


- 0-27 


0-0729 


16-01 


- 0-21 


0-0441 


15-95 


- 0-27 ! 0-0729 


15-96 


- 0-26 


0-0676 


15-91 


- 0-31 


0-0961 


15-88 


- 0-:i4 


0-1156 


15-88 


- 0-34 


0-1156 


15-93 


- 0-29 


0-0841 


15-91 


- 0-31 


0-0961 








Mean of 13 observations = 16-22 ; 2(x 2 ) = 13-9659. 



The deviation of the suspected observation from the mean, is 3-59. By 
Chauvenet's criterion, probable error = r = -7281, n = 13. From Table XII., 
x\r = 3-07, .'. x = 3-07 x -7281 = 2-24. Since the observation 19-81 deviates 
from the mean more than the limit 2-24 allowed by Chauvenet's criterion, 
that observation must be rejected. 

(2) Should 16-01 be rejected from the preceding set of observations? 
Treat the twelve remaining after the rejection of 19-81 exactly as above. 

(3) Should the observations 0-3902 and 0-3840 in Rudberg's results, 
page 441, be retained ? 

(4) Do you think 203-666 in Crookes' data, page 445, is affected by some 
" mistake " ? 

(5) Would Rowland have rejected the " 442-8 " result in Joule's work, 
page 4*T, if he had been solely guided by Chauvenet's criterion ? 

(6) Some think that "4'88" in Cavendish's data, page 466", is a mistake. 
Would you reject this number if guided by the above criterion ? 

These examples are given to illustrate the method of applying the criterion. 
Nothing more. Any attempt to establish an arbitrary criterion applicable to 
all cases, by eliminating the knowledge of the investigator, must prove un- 
satisfactory. It is very questionable if there can be a better guide than the 
unbiassed judgment and common-sense of the investigator himself. 

Any observation set aside by reason of its failure to comply 
with any test should always be recorded. As a matter of fact, the 
rare occurrence of abnormal results serves only to strengthen the 
theory of errors developed from the empirical formula, y = ke ~ * * 
There can be no doubt that as many positive as negative chance 
deviations would appear if a sufficient number of measurements 
were available.* " Every observation," says Gerling in his Die 
Ausgleichungs-Rechnungen der praktischen Geometric (Hamburg, 68, 
1843), " suspected by the observer is to me a witness of its truth. 



* Edgeworth has an interesting paper 
Mag. [5], 23, 364, 1887. 



On Discordant Observations " in the Phil* 



478 HIGHER MATHEMATICS. $ 187. 

He has no more right to suppress its evidence under the pretence that 
it vitiates the other observations than he has to shape it into con- 
formity with the majority." The whole theory of errors is founded 
on the supposition that a sufficiently large number of observations 
has been made to locate the errors to which the measurements 
are susceptible. When this condition is not fulfilled, the abnormal 
measurement, if allowed to remain, would exercise a dispropor- 
tionate influence on the mean. The result would then be less 
accurate than if the abnormal deviation had been rejected. The 
employment of the above criterion is, therefore, permitted solely 
because of the narrow limit to the number of observations. It is 
true that some good observations may be so lost, but that is the 
price paid to get rid of serious mistakes. 

It is perhaps needless to point out that a suspected observation 
may ultimately prove to be a real exception requiring further 
research. To ignore such a result is to reject the clue to a new 
truth. The trouble Lord Eayleigh recently had with the density of 
nitrogen prepared from ammonia is now history. The " ammonia " 
nitrogen was found to be i^oo^h P ar * lighter than that obtained 
from atmospheric air. Instead of putting this minute " error " on 
one side as a "suspect," Lord Eayleigh persistently emphasised 
the discrepancy, and thus opened the way for the brilliant work of 
Eamsay and Travers on " Argon and Its Companions ". 



179 



CHAPTER XII. 
COLLECTION OF FORMULAE FOR REFERENCE. 

188. Laws of Indices and Logarithms. 

THE average student of chemical science is compelled to take a 
course in pure mathematics. But after passing his " intermediate," 
all is forgotten except a strong prejudice that mathematics is a 
compilation of vexatious puzzles. This is to be regretted, because 
with very little, if any, more drilling the later chapters of mathe- 
matics would be found invaluable auxiliaries in the inquiry into 
those very phenomena to which he subsequently devotes his 
attention. 

Certain sections of this chapter have been written to give the 
student of this work the opportunity of revising some of the more 
fundamental principles established in elementary mathematics ; 
other sections are only for reference upon special occasions. 

To continue the discussion opened at the commencement of 
16, page 34, 

4x4= 16, is the second power of 4, written 4- ; 
4x4x4= 64, is the third power of 4, written 4 3 ; 
4x4x4x4 = 256, is the fourth poiver of 4, written 4 4 ; 

and in general, the nth power of any number a, is denned as the 
continued product 

a x a x a x ... n times = a", 

where n is called the exponent or index of the power. 
By actual multiplication, therefore, 

10 2 x 10 s = 10 2 + 3 = HP = 100,000 ; 
or, in general symbols, 

a m x a" = a'* + " ; or, a' x a" x a : x . . . = a { * + '+ ', 
a result known as the index law. Again, 

3x5 = (10 ' 4 " 1 ) x (lO ' 0990 ) = 10 1 ' 1701 - 15, 
because, from a table of common logarithms, 

Iog 10 3 = 0-4771 ; lo gl(J 5 = 0-6990; Iog 10 15 = M7UL 



480 HIGHER MATHEMATICS. 188. 

Thus we have performed arithmetical multiplication by the simple 
addition of two logarithms. To generalise : 

To multiply two or more numbers, add the logarithms of 
the numbers and find the number whose logarithm is the sum of the 
logarithms just obtained. 

EXAMPLES. (1) Evaluate 4 x 80. 

Iog 10 4 = 0-6021 
Iog 10 80 = 1-9031 



Sum = 2-5052 = Iog 10 320. 
.-. Ansr. = 320. 

This method of calculation holds good whatever numbers we employ in place- 
of 3 and 5 or 4 and 80. Hence the use of logarithms for facilitating numerical 
calculations. We shall shortly show how the operations of division, involu- 
tion, and evolution are as easily performed as the above multiplication. 

(2) Show logee = 1, log e l = 0. 

Just as 1 = 10, 2 = 10 ' 301 , 3 = 10' 477 , . . . ; 

so is 1 = c, 2 = e- 69 -' 2 , 3 - e 1>W86 , . . . ; 

where e 2-71828 . . . Hence by the definition of logarithms, 

Iog e 3 = 1-0986 ; Iog e 2 = 0-6932 ; log e l = 0. 
Again 

e x e x e x . . . n times = e n ; . . . ; e x e x e = e* ; e x e = e 2 ; e = e l ; 
or loge n = n\ . . . ; logeC :J = 3 ; loget' 2 = 2 ; log^ 1 = 1 = 



From the above it also follows that 



* -2 



= 10 ; or generally, . 



Hence the rule : 

To divide two numbers, subtract the logarithm of the divisor 
(denominator of a fraction) from the logarithm of the dividend 
(numerator of a fraction) and find the number corresponding to the 
resulting logarithm. 

EXAMPLES. (1) Evaluate 60 4- 3. 

Iog in 60 = 1-7782 



Difference = 1-3011 = Iog 10 20. 

Ansr. = 20. 
(2) Show that 2 ~ 2 = ; 10 - 2 = T ^ ; 3* x 3 ~ 3 = 1. 

It is very easy to miss the meaning of the so-called " properties 
of indices," unless the general symbols of the textbooks are 
thoroughly tested by translation into numerical examples. The 
majority of students require a good bit of practice before a general 



188. COLLECTION OF FORMULA I. I-ni; REFERENCE, 481 

expression * appeals to them with full force. Here, as elsewhere, 
it is not merely necessary for the student to think that he " under- 
stands the principle of the thing," he must actually work out 
examples for himself. " In scientiis ediscendis prosunt exempla 
magis quam prsecepta " f is as true to-day as it was in Newton's 
time. For example, how many realise why mathematicians write 
e = 1, until some such illustration as the following has been 
worked out? 

22 x 2 = 2 2 + = 2 2 = 4. 

The same result, therefore, is obtained whether we multiply 2 2 by 
2 or by 1, i.e., 

2 2 x 2 = 2 2 x 1 = 2 2 = 4. 

Hence it is inferred that 

2 = 1, and generally that a = l.J 

I am purposely using the simplest of illustrations, leaving the 
reader to set himself more complicated numbers. No pretence is 
made to rigorous demonstration. We assume that what is true in 
one case, is true in another. It is only by so collecting our facts 
one by one that we are able to build up a general idea. The be- 
ginner should always satisfy himself of the truth of any abstract 
principle or general formula by applying it to particular and simple 
cases. 

By actual multiplication show that 

(100) 3 = (10 2 ) 8 = 10 2X3 = 10 6 , 
and hence : 

To raise a number to any power, multiply the logarithm of 
the number by the index of the power and find the number corre* 
spending to the resulting logarithm. 



* The general symbols a, b, . . . m, n, . . . x, y, . . . in any general expression 
may be compared with the blank spaces in a bank cheque waiting to have particular 
values assigned to date, amount ( s. d.), and sponsor, before the cheque can fulfil the 
specific purpose for which it was designed. So must the symbols, a, 6, ... of a 
general equation be replaced by special numerical values before the equation can be 
applied to any specific process or operation. 

f Which may be rendered: "In learning we profit more by example than by 
precept". 

$ Some mathematicians define a'aslxaxaxa. . . n times ; a' = 1 x a x a x a ; 
a 2 = 1 xaxa;a 1 = lxa; and a as 1 x a no times, that is unity itself. If so, then 
would mean 1 x no times, i.e., 1 ; 1/0 would mean 1/(1 x no times), i.e., unity. 
But see examples, 5. 

HH 



482 HIGHER MATHEMATICS. 188. 

EXAMPLE. Evaluate 5 2 . 

o 2 = (5) 2 = (10 If5990 ) 2 = 10 1 ' 3980 = 25, 

since reference to a table of common logarithms shows that 
Iog 10 5 = 0-6990 ; Iog 10 25 = 1-3980. 

From the index law, above 

10* x 10* = 10* + * = 10 1 = 10. 

That is to say, 10* multiplied by itself gives 10. But this is the 
definition of the square root of 10. 

.-. ( VK))2 = VlO x x/10 = 10* x 10* = 10. 

A fractional index, therefore, represents a root of the particular 
number affected with that exponent. Generalising this idea, the 

nth root of any number a, is a. Thus 

#8 = 8*, because /8 x ?/8 x ^8 = 8* x 8* x 8* = 8. 
To extract the root of any number, divide the logarithm of 
the number by the index of the required root and find the number 
corresponding to the resulting logarithm. 

EXAMPLES. (1) Evaluate 4/8 and ^93". 

4/8 = (8p = (I0 >9031 p = 10 ' 3010 = 2 ; 
v 93 = (93)* = (I0 1>968 y = 10 ' 2812 = 1-91, 



'since, from a table of common logarithms, 

Iog 10 2 = 0-3010 ; Iog 10 8 = 0-9031 ; Iog 10 l-91 = 0-2812 ; Iog 10 93 = 1-9685. 

(2) Repeat all the above illustrations of the index law using Table XXIV., 
page 520. 

The results of logarithmic calculations are seldom absolutely 
correct because we employ approximate values of the logarithms 
of the particular numbers concerned. Instead of using logarithms 
to four decimal places we could, if stupid enough, use logarithms 
accurate to sixty-four decimal places. But this question is reserved 
for the next section. 

The more important properties of indices known under the name 
" the theory of indices " are summarised in the subjoined synopsis 
along with the corresponding properties of logarithms. 



$189. COLLECTION OK !( >li.M T I. A K l-m; REFERENCE 483 



Theory of Indices. 


Logaritlnu.. 




a=l. . . ' . 


log 1 = 


(1) 




\r\ff ri 1 




a* Wa 
a n = a x a x a x ... n times. . 


log^Va)}log^if 

log a" = 7^1oga. . . . 


(3) 


a= oo, if a > 1. 


log oo = oo 


'. (5) 


2 : =r/a.". : : : : 


log(-l)=0. . . . 
logj./a = - l log,, = - 1. . 


: l?i 


a~ n I/a 11 . 


loff n ~ n 7? 1n0 n w 


in 




log I/ \/a = - $ log a a = - J. 


IP) 

. (9) 


a-" = 0, if a > 1. 


log = - oo. 


. (10) 


fl W ft n * = tt( n + m ). .... 


log & = log a + log 6. 


. (11) 


o n 6" = (a&) 


Iog(a6) n = n log a + n log 6. 


. (12) 


\f'2/a=' n ya 


log'^a^^-logrt. . 


. (13) 


<'*/& -tfrt 


log \ ! ab = -log a + -log ft. 


. (14) 


a*/a" = a("- OT ) 


log a/6 = log a - log 6. 


(15) 


(a")' = a" 1 " 


log a" = ?^ log a. . 


. (16) 


3^ =a' =(#)" 


logVa = ^loga. . 


(17) 



EXAMPLE. Plot log e x = y, and show that logarithms of negative numbers 
are impossible. Hint, put x = 0, 1/e 2 , l/e, 1, e, e 2 , oo, etc., and find corresponding 
values of y. 

NOTE. Continental writers variously use the symbols L, /, In, Ig, for 
"log"; and "log nep " or "log nat " for " log*". "Nep" is an abbreviation 
for "Neperian," a Latinized adjectival form of Napier's name. 

" Exp x " is sometimes written for " e* " ; " Exp( - x) " for " c . 



189. Approximate Calculations in Scientific Work. 

A good deal of the tedious labour involved in the reduction of 
experimental results to their final form, may be avoided by atten- 
tion to the degree of accuracy of the measurements under con- 
sideration. It is one of the commonest of mistakes to. extend the 
arithmetical work beyond the degree of precision attained in the 
practical work.* Thus, Dulong calculated his indices of refraction 
to eight digits when they agreed only to three. When asked 
" Why?", Dulong returned the ironical answer : " I see no reason 

* In a memoir " On the Atomic Weight of Aluminum," at present before me, I 
read, " -646 grm. of aluminum chloride gave 2-0549731 grnis. of silver chloride . . .". 
It is not clear how the author obtained his seven decimals seeing that, in an earli.-i 
part of the paper, he expressly states that his l>aluiu'i' was not sensitive to more than 
0001 grm. 



484 HIGHER MATHEMATICS. 189. 

for suppressing the last decimals, for, if the first are wrong, the 
last may be all right " ! 

Although the measurements of a Stas, or of -a Whitworth may 
require six or eight decimal figures, few observations are correct 
to more than four or five. But even this degree of accuracy is 
only obtained by picked men working under special conditions. 
Observations which agree to the second or third decimal place are 
comparatively rare in chemistry. 

Again, the best of calculations is a more or less crude approxi- 
mation on account of the ''simplifying assumptions" introduced 
when deducing the formula to which the experimental results 
are referred. It is, therefore, no good extending the " calculated 
results " beyond the reach of experimental verification. It is un- 
profitable to demand a greater degree of precision from the calcu- 
lated than from the observed results but one ought not to demand 
a less. (Compare the introduction to Poincare's Mecanique Celeste.) 

The general rule in scientific calculations is to use one more 
decimal figure than the degree of accuracy of the data. In other 
words, reject as superfluous all decimal figures beyond the. first 
doubtful digit. The remaining digits are said to be significant 
figures. 

EXAMPLES. In 1-540, there are four significant figures, the cypher indi- 
cates that the magnitude has been measured to the thousandth part ; in 
0-00154, there are three significant figures, the cyphers are added to fix the 
decimal point; in 15,400, there is nothing to show whether the last two 
cyphers are significant or not, there may be three, four, or five significant 
figures. 

In " casting off" useless decimal figures, the last digit retained 
must be increased by unity when the following digit is greater 
than four. We must, therefore, distinguish between 9-2 when 
it means exactly 9*2, and when it means anything between 9'14 
and 9-25. In the so-called " exact sciences," the latter is the 
usual interpretation. Quantities are assumed to be equal when 
the differences fall within the limits of experimental error. 

LOGARITHMS. There are very few calculations in practical work outside 
the range of four or five figure logarithms. The use of more elaborate tables 
may, therefore, be dispensed with.* 



* Thus Wrapson and Gee's Mathematical Tables (Is. 6d. ) to four decimal places 
may be used instead of Chamber's (page 37) to seven decimal places. 



$189. COLLECTION OF FORMULA!: Fn|; KKPERENn- I8B 

ADDITION AND SUBTRACTION. In adding .such numbers as 9-2 and 0-4918, 
cast off the 8 and the 1, then write the answer, 9-69, not 9-6913. Show that 
5-60 + 20-7 + 103-193 = 129-5, with an error of about 0-01, that is about 0-08 / . 

MULTIPLICATION AND DIVISION. The product 2-25ir represents the length 
of the perimeter of a circle whose diameter is 2-25 units; * is a numerical 
coefficient whose value has been calculated by Shanks,* to over seven hnmlrr.l 
decimal places, so that it = 3-141592,653589,793. ... Of these two numbers, 
therefore, 2-25 is the less reliable. Instead of the ludicrous 7-0685808625 . . ., 
we simply write the answer, 7*07. 

It is no doubt unnecessary to remind the reader that in scientific compu- 
tations the standard arithmetical methods of multiplication and division 
are abbreviated so as to avoid writing down a greater number of digits than 
is necessary to obtain the desired degree of accuracy. The following scheme 
for " shortened multiplication and division," requires little or no explanation : 

Shortened Multiplication. Shortened Division. 

9-774 365-4)3571-3(9-774 

365-4 3288-6 



2932-2 282-7 

586-4 255-8 

48-9 

3 ' 9 25-5 



3571-4 1>4 

The digits of the multiplier are taken from left to right, not right to left. 
One figure less of the divisor is used at each step of the division. The last 
figure of the quotient is obtained mentally. A "bar" is usually placed over 
strengthened figures so as to allow for an excess or defect of them in the 
result. 

Ostwald, in his Hand- mid Hilfsbuch zur Aus/iihruwj physiko- 
chemiker Messungen (Leipzig, 1893), has said that " the use of 
these methods cannot be too strongly emphasised. The ordinary 
methods of multiplication and division must be termed unscientific." 
Full details are given in Langley's booklet A Treatise on Compu- 
tation (Longmans, Green & Co., 1895), or in the more formal 
Calculs pratiques appliques aux Sciences d' Observation, by Babinet 
and Housel. 

The error introduced in approximate calculations by the " caxtimj 
off" f decimal figures. Some care is required in rounding off 
decimals to avoid an excess or defect of strengthened figures by 
making the positive and negative errors neutralise each other in 
the final result. A good " dodge " is always to leave the last figure 

A'..//. Soc., 22,4:.. 1873. 



486 HIGHER MATHEMATICS. 189. 

an even number. E.g., 3 '75 would become 3*8, while 3 - 85 would 
be written 3*8. 

The percentage error of the product of two approximate numbers 
is very nearly the algebraic sum of the percentage error of each. 
If the positive error in the one be numerically equal to the negative 
error in the other, the product will be nearly correct, the errors 
neutralise each other. 

EXAMPLE. 19-8 x 3'18. The first factor may be written 20 with a + 
error of 1 / , and, therefore, 20 x 3-18 = 63-6, with a + error of 1 / . This 
excess must be deducted from 63-6. We thus obtain 62-95. The true result is 
62-964. 

The percentage error of the quotient of two approximate numbers 
is obtained by subtracting the percentage error of the numerator 
from that of the denominator. If the positive error of the nume- 
rator is numerically equal to the positive error of the denominator, 
the error in the quotient is practically neutralised. 

Vide footnote, page 454. 

APPROXIMATION FORMULAE CALCULATIONS WITH SMALL QUAN- 
TITIES. The discussion on approximate calculations in Chapter V. 
renders any further remarks on the deduction of the following 
formulae superfluous : 

For the sign of equality, read " is approximately equal to," or " is very 
nearly equal to ". Let a, /8, 7, . . . be small fractions in comparison with 
unity or x. 

(1 a) (1 j8) = 1 a j8 ........ (1) 

(1 ) (1 0) (1 7) = 1 ft 7 ..... ( 2 ) 
(1 a) 2 = 1 2a ; (1 a) H = 1 na ...... (3) 

V(l + ) = 1 + * * / * = K + 0) ...... (*) 



) = /(l a) 



The third member of some of the following results is to be regarded as a 
second approximation, to be employed only when an exceptional degree of 
accuracy is required. 

e* = 1 -f a ; a = 1 + a log a ........ (7) 

log (1 + a) = a = a - Aa 2 ........ (8) 

log (x + a) = log x + a/X - ia 2 /C 2 ....... (9) 

x + a 2a 2 a 2 

"w^r-.-F + s-S ......... 

By Taylor's theorem, 99, 

sin (x + )8) = sin x + cos x - ^0 2 sin x 



190. COLLECTION OF FORMULA I! l-'n|; REFERENCE 487 

If the angle is not greater than 2$, /3 < -044 ; J/8* < -001 ; J/F < -00001. 

But sin x does not exceed unity, therefore, we may look upon 
sin(x + /8) = sinx + ft cos x, 

correct up to three decimal places. The addition of another term "- 4/8*" 

will make the resul